Broadband polarisation independent photonic switch devices and methods

Abstract

Optical networking has become an enabling technology for the Internet and telecommunication networks that transmit the vast majority of all human and machine-to-machine information today. Such networks exploit photonic interconnection architectures to address multiple conflicting demands or route traffic etc. Photonic integrated circuit devices offer smaller footprint, reduced power consumption, reduced latency, higher interconnection counts and lower cost for large photonic switch fabrics employing space and / or wavelength domains with fast or slow switching speeds. Such fabrics should exploit broadband, polarization independent and high extinction-ratio optical switch elements that provide the required performance, scalability, and cost balance. The inventors have established complex Mach-Zehnder-based switches with large width variable cross-section width waveguides comprising 1x2 and 2x2 splitting stages to achieve these broadband, polarization independent and high extinction-ratio optical switch elements with high performance and wide manufacturing tolerances.

Description

BROADBAND POLARISATION INDEPENDENT PHOTONIC SWITCH DEVICES AND METHODS CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This patent application claims the benefit of priority to U.S. Provisional Patent Application 63 / 582,594 filed September 14, 2023, the entire contents of which are incorporated herein by reference. FIELD OF THE INVENTION

[0002] This patent application relates to photonic switching and more particularly to broadband, polarization independent high extinction ratio photonic switching devices and methods for designing broadband, polarization independent high extinction ratio photonic switches using large variable cross-section waveguides. BACKGROUND OF THE INVENTION

[0003] Optical networking is a means of communication that uses signals encoded in light to transmit information in various types of telecommunications networks. These include limited range local-area networks (LAN) or wide-area networks (WAN), which cross metropolitan and regional areas as well as long-distance national, international and transoceanic networks. Optical networks typically employ optical amplifiers, lasers, modulators, optical switches and wavelength division multiplexing (WDM) to transmit large quantities of data, generally across fiber-optic cables. Because it is capable of achieving extremely high bandwidth, it is an enabling technology for the Internet and telecommunication networks that transmit the vast majority of all human and machine-to-machine information today. Optical networks are also employed in other applications such as storage area networks and data centers.

[0004] For example, within data centers there is a requirement to provide reconfigurable connectivity at multiple levels such as chip-to-chip, server-to-server, rack-to-rack, and cluster- to-cluster exploiting photonic interconnection architectures to address the multiple conflicting demands. It would be further beneficial to exploit photonic integrated circuit devices that support large photonic switch fabrics employing space and / or wavelength domains with either fast or slow switching speeds to provide distributed photonic switch fabrics employing combinations of fast and slow photonic switching elements allowing these to provide reducedlatency, increased flexibility, lower cost, lower power consumption, and high interconnection counts.

[0005] It would be beneficial for such photonic switches to exploit broadband, polarization independent and high extinction-ratio optical switch elements that provide the required performance, scalability, and cost balance to support the establishment of optical switch blocks and matrices to support network deployments within a range of applications.

[0006] Within embodiments of the invention fast switching (via electro-optic effects for example) or slow switching (via thermos-optic effect for example) switch matrices employing complex Mach-Zehnder-based switches with large width variable cross-section width waveguides comprising 1x2 and 2x2 splitting stages to achieve a broadband, polarization independent and high extinction-ratio optical switch elements with high performance and wide manufacturing tolerances.

[0007] Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. SUMMARY OF THE INVENTION

[0008] It is an object of the present invention to mitigate limitations in the prior art relating to photonic switching and more particularly to broadband, polarization independent high extinction ratio photonic switching devices and methods for designing broadband, polarization independent high extinction ratio photonic switches using large variable cross-section waveguides.

[0009] .In accordance with an embodiment of the invention there is provided a method comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range; establishing a waveguide geometry wherein a polarization phase difference within an active switching section of the interferometric optical switch comprising a difference between a phase for optical signals in a transverse magnetic (TM) mode of an optical waveguide forming the active section and another phase for optical signals in a transverse electric (TE) mode of the optical waveguide forming the active section is below a predetermined threshold over a defined range of phase differences between optical signals propagating within an arm of the interferometric optical switch wherein thedefined range of phase differences is at least that to change a switch state of the interferometric optical switch to another switch state of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides.

[0010] In accordance with an embodiment of the invention there is provided a method comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range; establishing a stability condition for the optical waveguides; establishing a waveguide geometry fulfilling the stability condition for an active switching section of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides. the stability condition is defined as a ratio of first rate of change to a second rate of change; the first rate of change is a differential of an effective index of a transverse magnetic (TM) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both a width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and a temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch; and the second rate of change is differential of an effective index of a transverse electric (TE) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both the width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and the temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch.

[0011] In accordance with an embodiment of the invention there is provided an interferometric optical switch comprising: an input optical coupler for coupling optical signals with a defined wavelength range from an input port to an end of each optical waveguide of a pair of optical waveguides; an output optical coupler coupled to a distal end of optical waveguide of the pair of optical waveguides for coupling the optical signals to an output port or another output port; and the pair of optical waveguides; wherein each optical waveguide of the pair of optical waveguides comprises a first section of a first length and a first width and a second section of a second length and a second width; the first length and the second length satisfy a condition that a first summation equals a second summation; the first summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse magnetic (TM) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TM mode of the second section of the optical waveguide; and the second summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse electric (TE) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TE mode of the second section of the optical waveguide.

[0012] In accordance with an embodiment of the invention there is provided a method comprising: providing a first 1x2 interferometric optical switch; providing a second 1x2 interferometric optical switch; providing a first 2x1 interferometric optical switch; and providing a second 2x1 interferometric optical switch; wherein a first output of the first 1x2 interferometric optical switch is coupled to an input of the first 2x1 interferometric optical switch; a second output of the first 1x2 interferometric optical switch is coupled to an input of the second 2x1 interferometric optical switch; a first output of the second 1x2 interferometric optical switch is coupled to another input of the first 2x1 interferometric optical switch; anda second output of the second 1x2 interferometric optical switch is coupled to another input of the second 2x1 interferometric optical switch.

[0013] Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Embodiments of the present invention will now be described, by way of example only, with reference to the attached Figures, wherein:

[0015] Figure 1 depicts a schematic of a 2x2 Mach-Zehnder interferometer (MZI) with input ports A and B, output ports C and D, arm 1 (wg1), arm 2 (wg2), input coupler (multimode interference coupler (MMI) 1), and output coupler (MMI 2);

[0016] Figure 2 depicts the switch extinction ratio in cross state versus couplers average coupling ratio, for different values of coupler coupling ratio difference;

[0017] Figure 3 depicts switch extinction ratio in bar state versus couplers coupling ratio difference, for different values of couplers average coupling ratio;

[0018] Figure 4 depicts a schematic of a 1x2 MZI schematic with input port A, output ports C and D, arm 1 (wg1), arm 2 (wg2), input coupler (MMI 1), and output coupler (MMI 2);

[0019] Figure 5 depicts waveguide effective index and group index versus waveguide width;

[0020] Figure 6 depicts a derivative over width of waveguide effective index and group index, versus width;

[0021] Figure 7 depicts a derivative over temperature of waveguide effective index and group index versus width;

[0022] Figure 8 depicts a stability coefficient ^^ versus waveguide width;

[0023] Figures 9A and 9B depict the transmission spectrum of 2x2 switch (bar state) for a waveguide width w0 of 246 nm, with waveguide width difference W of 0 nm and 2.0 nm respectively;

[0024] Figures 10A and 10B depict the transmission spectrum of 2x2 switch (bar state) for a waveguide width w0 of 450 nm, with a waveguide width difference W of 0 nm and 2.0 nm respectively;

[0025] Figures 11A and 11B depict the transmission spectrum of 2x2 switch (bar state) for a waveguide width w0 of 700 nm, with a waveguide width difference W of 0 nm and 2.0 nm respectively;

[0026] Figures 12A and 12B depict the transmission spectrum of 2x2 switch (bar state) for a waveguide width w0 of 2500 nm, with a waveguide width difference W of 0 nm and 2.0 nm respectively;

[0027] Figure 13A and 13B depict the transmission spectrum of 2x2 switch (cross state) for a waveguide width w0 of 246 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0028] Figures 14A and 14B depict the transmission spectrum of 2x2 switch (cross state) for a waveguide width w0 of 450 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0029] Figures 15A and 15B depict the transmission spectrum of 2x2 switch (cross state) for a waveguide width w0 of 700 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0030] Figures 16A and 16B depict the transmission spectrum of 2x2 switch (cross state) for a waveguide width w0 of 2500 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0031] Figure 17A and 17B depict the extinction ratio of 2x2 switch versus temperature difference for a waveguide width w0 of 246 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0032] Figures 18A and 18B depict the extinction ratio of 2x2 switch versus temperature difference for a waveguide width w0 of 450 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0033] Figures 19A and 19B depicts the extinction ratio of 2x2 switch versus temperature difference for a waveguide width w0 of 700 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0034] Figures 20A and 20B depict the extinction ratio of 2x2 switch versus temperature difference for a waveguide width w0 of 2500 nm, with a waveguide width difference W of 0 nm and 2.0 nm;

[0035] Figures 21A, 21B, 21C and 21D depict the free spectral range (FSR) of switch versus waveguide width difference W for different target phase differences ϕs (corresponding to specific switch states), for waveguide widths of 246 nm, 450 nm, 700 nm, and 2500 nm;

[0036] Figure 22 depicts a stability coefficient ^^, ^ versus waveguide width;

[0037] Figure 23 depicts a stability coefficient ^^, ^^ versus waveguide widths ^^ and ^^;

[0038] Figure 24 depicts a length ratio versus waveguide widths ^^ and ^^;

[0039] Figures 25A and 25B depict a stability coefficient ^^, ^^ and length ratio ^^ / ^^versus waveguide width ^^ for ^^ = 2500 nm;

[0040] Figures 26A and 26B depict transmission spectra of a 2x2 switch tuned at cross state and bar state for waveguide widths ^^ = 2500 nm and ^^ = 275 nm, with a waveguide width difference ^ = 2 nm and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0041] Figures 27Aand 27B depict transmission spectra of a 2x2 switch tuned at cross state and bar state for waveguide widths ^^ = 2500 nm and ^^ = 265 nm, with a waveguide width difference ^ = 2 nm and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0042] Figures 28A and 28B depict transmission spectra of a 2x2 switch tuned at cross state and bar state for waveguide widths ^^ = 2500 nm and ^^ = 285 nm, with a waveguide width difference ^ = 2 nm and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0043] Figure 29 depicts an extinction ratio of a 2x2 switch versus temperature difference for waveguide widths ^^ = 2500 nm and ^^ = 275 nm, with a waveguide width difference ^ = 2 nm and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0044] Figure 30 depicts a stability coefficient ^^, ^ versus waveguide width;

[0045] Figure 31 depicts a 2x2 quad-switch switch comprises four sub-switches 1×2 (dotted line, as in Figure 4), each using a coupler 1×2 and a coupler 2×2;

[0046] Figure 32 depicts 2x2 quad-switch switch of Figure 31 in cross state;

[0047] Figure 33 depicts a 2x2 quad-switch switch of Figure 31 in bar state;

[0048] Figure 34A and 34B depict transmission spectra of a 2x2 switch tuned at cross state and bar states with a waveguide width difference W = 0 and MMI coupling ratios κ1=κ2=0.5;

[0049] Figure 35A and 35B depict transmission spectra of a 2x2 switch tuned at cross state and bar state with a waveguide width difference W = 2 nm and MMI coupling ratios κ1=κ2=0.5;

[0050] Figure 36A and 36B depict transmission spectra of a 2x2 switch tuned at cross state and bar state with a waveguide width difference W = 0 and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0051] Figures 37A and 37B depict transmission spectra of 2x2 switch tuned at cross state and bar state with a waveguide width difference W = 2 nm and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0052] Figures 37C and 37D depict zoomed portions of the main output of the 2x2 switch as depicted in Figures 37A and 37B;

[0053] Figures 38A and 38B depict transmission spectrum of a 2x2 quad-switch tuned at cross state and bar state with a waveguide width difference W = 0 and MMI coupling ratios κ1=κ2=0.5;

[0054] Figures 39A and 39B depict transmission spectrum of a 2x2 quad-switch tuned at cross state and bar state with a waveguide width difference W = 2 nm and MMI coupling ratios κ1=κ2=0.5;

[0055] Figures 40A and 40B depict transmission spectrum of a 2x2 quad-switch tuned at cross state and bar state with a waveguide width difference W = 0 and MMI coupling ratios κ1=0.5 and κ2=0.515;

[0056] Figures 41A and 41B depict transmission spectrum of a 2x2 quad-switch tuned at cross state and bar state with a waveguide width difference W = 2 nm and MMI coupling ratios κ1=0.5 and κ2=0.515;

[0057] Figures 41C and 41D depict zoomed portions of the main output of the 2x2 switch as depicted in Figures 40A and 40B;

[0058] Figure 42 depicts an extinction ratio of 2x2 switch versus temperature difference (waveguide width w0=2500 nm, waveguide width difference W = 2 nm, and MMI coupling ratios κ1=0.51 and κ2=0.52);

[0059] Figure 43 depicts an extinction ratio of 2x2 quad-switch versus temperature difference (waveguide width w0=2500 nm, waveguide width difference W = 2 nm, and MMI coupling ratios κ1=0.5 and κ2=0.515);

[0060] Figure 44 depicts the FSR of switch versus waveguide width difference W for different target phase differences ϕs (corresponding to specific switch states), for a waveguide width of 2500 nm;

[0061] Figures 45A and 45B depict transmission spectrum of a 2x2 switch tuned at 4% near cross state and 4% near bar state with waveguide width ^0 = 2500 nm, waveguide width difference W = 2 nm, and MMI coupling ratios κ1=0.51 and κ2=0.52;

[0062] Figure 46A and 46B depict transmission spectrum of a 2x2 quad-switch tuned at 4% near cross state and 4% near bar state with waveguide width ^0 = 2500 nm, waveguide width difference W = 2 nm, and MMI coupling ratios κ1=0.5 and κ2=0.515;

[0063] Figure 47 depicts a configuration for a 2x2 quad-switch without waveguide crossings according to embodiments of the invention;

[0064] Figure 48 depicts 2x2 octo-switch employing eight 1x2 sub-switches according to embodiments of the invention;

[0065] Figure 49 depicts a 2x2 dodeca-switch employing twelve 1x2 sub-switches according to embodiments of the invention; and

[0066] Figure 50 A 1x2 tri-switch employing three 1x2 sub-switches according to embodiments of the invention.DETAILED DESCRIPTION

[0067] The present invention is directed to photonic switching and more particularly to broadband, polarization independent high extinction ratio photonic switching devices and methods for designing broadband, polarization independent high extinction ratio photonic switches using large variable cross-section waveguides.

[0068] The ensuing description provides representative embodiment(s) only, and is not intended to limit the scope, applicability or configuration of the disclosure. Rather, the ensuing description of the embodiment(s) will provide those skilled in the art with an enabling description for implementing an embodiment or embodiments of the invention. It would be understood by one of skill in the art that various changes can be made in the function and arrangement of elements without departing from the scope of the invention as set forth in the claims. Accordingly, an embodiment is an example or implementation of the inventions and not the sole implementation. Various appearances of “one embodiment,” “an embodiment” or “some embodiments” do not necessarily all refer to the same embodiments. Although various features of the invention may be described in the context of a single embodiment, the features may also be provided separately or in any suitable combination. Conversely, although the invention may be described herein in the context of separate embodiments for clarity, the invention can also be implemented in a single embodiment or any combination of embodiments.

[0069] Reference in the specification to “one embodiment,” “an embodiment,” “some embodiments” or “other embodiments” means that a particular feature, structure, or characteristic described in connection with the embodiments is included in at least one embodiment, but not necessarily all embodiments, of the invention. The phraseology and terminology employed herein is not to be construed as limiting but is for descriptive purposes only. It is to be understood that where the claims or specification refer to “a” or “an” element, such reference is not to be construed as there being only one of that element. It is to be understood that where the specification states that a component feature, structure, or characteristic “may,” “might,” “can” or “could” be included, that particular component, feature, structure, or characteristic is not required to be included.

[0070] Reference to terms such as “left,” “right,” “top,” “bottom”, “front” and “back” are intended for use in respect to the orientation of the particular feature, structure, or element within the figures depicting embodiments of the invention. It would be evident that suchdirectional terminology with respect to the actual use of a device has no specific meaning as the device can be employed in a multiplicity of orientations by the user or users.

[0071] Reference to terms “including,” “comprising,” “consisting” and grammatical variants thereof do not preclude the addition of one or more components, features, steps, integers or groups thereof and that the terms are not to be construed as specifying components, features, steps or integers. Likewise, the phrase “consisting essentially of,” and grammatical variants thereof, when used herein is not to be construed as excluding additional components, steps, features integers or groups thereof but rather that the additional features, integers, steps, components or groups thereof do not materially alter the basic and novel characteristics of the claimed composition, device or method. If the specification or claims refer to “an additional” element, that does not preclude there being more than one of the additional elements.

[0072] A “two-dimensional” waveguide, also referred to as a 2D waveguide or a planar waveguide, as used herein may refer to, but is not limited to, an optical waveguide supporting propagation of optical signals within a predetermined wavelength range which guides the optical signals vertically relative to a substrate upon which the 2D waveguide is formed but does not guide the optical signals laterally relative to the propagation direction of the optical signals within the 2D waveguide.

[0073] A “three-dimensional” waveguide, also referred to as a 3D waveguide, a channel waveguide, or simply waveguide as used herein may refer to, but is not limited to, an optical waveguide supporting propagation of optical signals within a predetermined wavelength range which guides the optical signals vertically relative to the substrate upon which the 3D waveguide is formed as well as laterally relative to the propagation direction of the optical signals within the 3D waveguide.

[0074] A “photonic integrated circuit” (PIC) as used herein may refer to, but is not limited to, the monolithic integration of multiple integrated optics devices into a circuit formed upon a common substrate providing an optical routing and processing functionality. The PIC is fabricated using processing techniques at a wafer level, e.g. CMOS manufacturing flows, MEMS processing flows, etc.

[0075] A “wavelength band” as used herein may refer to, but is not limited to, a defined wavelength range over which a device or system is either intended to operate or provide functionality. Such wavelength bands may be associated with a standard, such as a telecommunication standard for example, by the design of the optical component(s) / system(s) or by an informal reference as known to those of skill in the art. A wavelength band may refer to the “O-band” which as used herein refers to, but is not limited to, the wavelength range1260-1360nm. A wavelength band may refer to the “E-band” which as used herein refers to, but is not limited to, the wavelength range 1360-1460nm. A wavelength band may refer to the “C-band” which as used herein refers to, but is not limited to, the wavelength range 1530–1565 nm. A wavelength band may refer to the “S-band” which as used herein refers to, but is not limited to, the wavelength range 1460–1530 nm. A wavelength band may refer to the “L-band” which as used herein refers to, but is not limited to, the wavelength range 1565–1625 nm. A wavelength band may refer to the “U-band” which as used herein refers to, but is not limited to, the wavelength range 1625-1675nm.

[0076] Within the embodiments of the invention the inventors refer to the term “hybridly integrated.” This may, within some embodiments of the invention, refer to, but not be limited to, the “integration” of an optical element onto a substrate by attaching the optical element or another element physically integrated with the optical element to the substrate (platform) such that the optical element is retained in position. Such attachment means may include, but not be limited to, soldering, epoxy, van der Waals forces, electrostatic attachment, magnetic attachment, physical interlocking and friction. Accordingly, in these embodiments of the invention the optical element being hybridly integrated may be viewed as being implemented within a parallel manufacturing process to the other optical element(s) prior to being co- assembled. This parallel manufacturing process may employ one or more processes selected from the group comprising, but not limited to, liquid phase epitaxy (LPE), metal organic chemical vapor deposition (MOCVD), organometallic vapor-phase epitaxy (OMVPE), selective area epitaxy, an additive manufacturing process, a non-additive manufacturing process, crystal growth, doping, induced damage, etching, doping and deposition.

[0077] This may, within other embodiments of the invention, refer to, but not be limited to, the “integration” of an optical element onto a substrate using a different manufacturing methodology and / or techniques to those employed in forming other optical components upon the substrate. For example, this may employ an LPE process to form the other optical element upon the substate wherein the optical component upon the substrate was formed by MOCVD or vice-versa. Alternatively, both the optical component and other optical component may be formed using the same manufacturing methodology or a combination of manufacturing methodologies. These manufacturing methodologies may employ one or more processes selected from the group comprising, but not limited to, LPE, MOCVD, OMVPE, selective area epitaxy, an additive manufacturing process, a non-additive manufacturing process, crystal growth, doping, induced damage, etching, doping, deposition, an additive manufacturing process and a non-additive manufacturing process. Accordingly, in these embodiments of theinvention the optical element being hybridly integrated may be viewed as being implemented within one or more further processing stages of the same manufacturing process as the other optical element(s). However, in each instance the optical waveguide and / or optical component properties require that an optical interface is implemented between the optical waveguide and optical component in order to provide efficient optical coupling between one and the other.

[0078] Within the embodiments of the invention the inventors refer to the term “monolithically integrated.” This may, within some embodiments of the invention, refer to, but not be limited to, the “integration” of optical elements onto or within a substrate directly forming each optical component onto or within the substrate (platform). The manufacturing processes for the optical elements may be concurrent, plesiochronous or asynchronous. Each optical element may be formed from one or more processes selected from the group comprising, but not limited to, LPE, MOCVD, OMVPE, selective area epitaxy, an additive manufacturing process, a non-additive manufacturing process, crystal growth, doping, induced damage, etching, doping and deposition.

[0079] An optical element may employ one or more semiconductors grown using LPE, MOCVD, and OMVPE, for example. The one or more semiconductors may be selected from, but not limited to, group III-V semiconductors, group II-VI semiconductors, group IV semiconductors, and group IV-V-VI semiconductors. Examples of group III-V semiconductors may include AlP, AlN, AlGaSb, AlGaAs, AlGaInP, AlGaN, AlGaP, GaSb, GaAsP, GaAs, GaN, GaP, InAlAs, InAlP, InSb, InGaSb, InGaN, GaInAlAs, GaInAlN, GaInAsN, GaInAsP, GaInAs, GaInP, InN, InP, InAs, InAsSb, and AlInN. Examples of group II-VI semiconductors may include ZnSe, HgCdTe, ZnO, ZnS, and CdO. Examples of group IV Semiconductors may include Si, Ge, and strained silicon. A group IV-V-VI semiconductor may be GeSbTe.

[0080] Within embodiments of the invention the platform or substrate upon which the integration is performed may be a silicon substrate wherein the one or more optical waveguidesupon the platform exploit a silicon nitride core with silicon oxide upper and lower cladding, a^^^^ − ^^^^^ − ^^^^ waveguide structure. Alternatively, the one or more optical waveguidesmay employ a silicon core with silicon nitride upper and lower claddings. Optionally, the upper cladding may be omitted within other embodiments of the invention.

[0081] However, it would be evident that other optical waveguide structures may be employed including, but not limited to, silica-on-silicon, doped (e.g., germanium, Ge) silica core with undoped cladding, silicon oxynitride, polymer-on-silicon, or doped silicon waveguides for example. Additionally, other waveguide structures may be employed includingvertical and / or lateral waveguide tapers and forming microball lenses on the ends of the waveguides via laser and / or arc melting of the waveguide tip.

[0082] Further, whilst embodiments of the invention are described with respect to silicon-on-insulator (SOI) waveguides by way of example, e.g. ^^^^ − ^^^^^ − ^^^^; ^^^^ − ^^: ^^^^ −^^^^; or ^^ − ^^^^;, it would be evident that within other embodiments of the invention maybe employed to coupled passive waveguides to active semiconductor waveguides, such as indium phosphide (InP) or gallium arsenide (GaAs), e.g. a semiconductor optical amplifier (SOA), laser diode, etc. Optionally, an active semiconductor structure may be epitaxially grown onto a silicon IO-MEMS structure, epitaxially lifted off from a wafer and bonded to a silicon integrated optical microelectromechanical systems (IO-MEMS) structure, etc.

[0083] However, within other embodiments of the invention a variety of waveguide coupling structures coupling onto and / or from waveguides employing material systems that include, but not limited to, ^^^^− ^^^^^− ^^^^; ^^^^− ^^: ^^^^− ^^^^; ^^ − ^^^^; ion exchanged glass, ion implanted glass, polymeric waveguides, InGaAsP, GaAs, III-V materials, II-VI materials, , and optical fiber. Whilst primarily waveguide-waveguide systems have been described it would be evident to one skilled in the art that embodiments of the invention may be employed in aligning intermediate coupling optics, e.g., ball lenses, spherical lenses, graded refractive index (GRIN) lenses, etc. for free-space coupling into and / or from a waveguide device.

[0084] Further, whilst embodiments of the invention are described primarily with respect to a silicon substrate it would be evident that other substrates may be employed within other embodiments of the invention. These may include, but not be limited to, a semiconductor, a ceramic, a metal, an alloy, a glass, or a polymer.

[0085] A “ceramic” as used herein may refer to, but is not limited to, an inorganic, nonmetallic solid material comprising metal, non-metal or metalloid atoms primarily held in ionic and covalent bonds. Such ceramics may be crystalline materials such as oxide, nitride or carbide materials, elements such as carbon or silicon, and non-crystalline. Exemplary ceramics may include high temperature ceramics or high temperature co-fired ceramics such as alumina (Al2O3), zirconia (ZrO2), and aluminum nitride (AlN) or a low temperature cofired ceramic (LTCC). A LTCC may be formed from a glass – ceramic combination.

[0086] A “glass” as used herein may refer to, but is not limited to, a non-crystalline amorphous solid. A glass may be fused quartz, silica, a soda-lime glass, a borosilicate glass, a lead glass, an aluminosilicate glass for example. A glass may include other inorganic andorganic materials including metals, aluminates, phosphates, borates, chalcogenides, fluorides, germanates (glasses based on GeO2), tellurites (glasses based on TeO2), antimonates (glasses based on Sb2O3), arsenates (glasses based on As2O3), titanates (glasses based on TiO2), tantalates (glasses based on Ta2O5), nitrates, carbonates, plastics, and an acrylic.

[0087] Further, whilst the embodiments of the invention are described and depicted with respect to a waveguide employing a core embedded within a cladding, a so-called buried waveguide, it would be evident that other waveguide geometries such as rib waveguide, diffused waveguide, ridge or wire waveguide, strip-loaded waveguide, slot waveguide, and anti-resonant reflecting optical waveguide (ARROW waveguide), photonic crystal waveguide, suspended waveguide, alternating layer stack geometries, sub-wavelength grating (SWG) waveguides and augmented waveguides (e.g. ^^ − ^^^^− ^^^^^^^ ). Further, whilst the embodiments of the invention are described and depicted with respect to a step-index waveguide it would be evident that other waveguide geometries such as graded index and hybrid index (combining inverse-step index and graded index) may be employed.

[0088] Optical networking is a means of communication that uses signals encoded in light to transmit information in various types of telecommunications networks. These include limited range local-area networks (LAN) or wide-area networks (WAN), which cross metropolitan and regional areas as well as long-distance national, international and transoceanic networks. Optical networks typically employ optical amplifiers, lasers, modulators, optical switches and wavelength division multiplexing (WDM) to transmit large quantities of data, generally across fiber-optic cables. Because it is capable of achieving extremely high bandwidth, it is an enabling technology for the Internet and telecommunication networks that transmit the vast majority of all human and machine-to-machine information today. Optical networks are also employed in other applications such as storage area networks and data centers.

[0089] For example, within data centers there is a requirement to provide reconfigurable connectivity at multiple levels such as chip-to-chip, server-to-server, rack-to-rack, and cluster- to-cluster exploiting photonic interconnection architectures to address the multiple conflicting demands. It would be further beneficial to exploit photonic integrated circuit devices that support large photonic switch fabrics employing space and / or wavelength domains with either fast or slow switching speeds to provide distributed photonic switch fabrics employing combinations of fast and slow photonic switching elements allowing these to provide reduced latency, increased flexibility, lower cost, lower power consumption, and high interconnection counts.

[0090] It would be beneficial for such photonic switches to exploit broadband, polarization independent and high extinction-ratio optical switch elements that provide the required performance, scalability, and cost balance to support the establishment of optical switch blocks and matrices to support network deployments within a range of applications.

[0091] Within embodiments of the invention fast switching (via electro-optic effects for example) or slow switching (via thermos-optic effect for example) switch matrices employing complex Mach-Zehnder-based switches with large width variable cross-section width waveguides comprising 1x2 and 2x2 splitting stages to achieve a broadband, polarization independent and high extinction-ratio optical switch elements with high performance and wide manufacturing tolerances.

[0092] Within the following embodiments of the invention embodiments of the invention are described and depicted with respect to silicon nitride based integrated photonic switching elements with suspended waveguides although within other embodiments of the invention other material systems may be employed with suspended or non-suspended waveguides.

[0093] Within embodiments of the invention thermo-optic compensation is described, e.g., bias tuning, to provide for improved tolerance to address fabrication process variations, for example, although within other embodiments of the invention other means / mechanisms of compensation may be employed, such as electro-optic compensation for example.

[0094] Within embodiments of the invention thermo-optic tuning is employed for switch state actuation although within other embodiments of the invention other embodiments of the invention other means / mechanisms of switch state actuation may be employed, such as electro-optic actuation for example.

[0095] Embodiments of the invention exploit, what the inventors refer to as large or wide waveguides, i.e. waveguide widths of 2000 - 2500nm for example for the first and second waveguides (wg1 and wg2) within the active region of the Mach-Zehnder interferometer (MZI) to decrease the sensitivity to variations arising from the fabrication process whilst also providing increased tolerance to polarization, increased free spectral range (FSR), improved the repeatability of the required electrical power, voltage, etc. of the actuation means employed for controlling the switching state.

[0096] Embodiments of the invention exploit variable waveguide cross-sections, i.e. different portions of wg1 and wg2 have different widths, within the active region of the Mach-Zehnder interferometer (MZI) to further provide improved tolerance to variations within the fabrication process.

[0097] Embodiments of the invention are described and depicted below employing a combination of 1x2 and 2x2 multimode interferometers (MMIs) within a 1x2 MZI which is employed as 1x2 switch discretely or in combination to provide higher order switches which provide increased optical bandwidth, enhanced extinction ratio, decreased control signals (e.g. required electrical tuning power). Such designs also provide for a decrease in the overall maximum operating temperature where thermos-optic control is employed and allow for a high extinction “off” state.

[0098] Within the following description the exemplary photonic switching devices are presented by way of a design methodology. It would be evident that the methodology of designing the optical switches may itself be considered an embodiment of the invention.

[0099] 1. SWITCH – 2X2 MACH-ZEHNDER INTERFEROMETER (MZI)

[0100] Transmission

[0101] The calculation of the transmission from a 2x2 Mach-Zehnder interferometer (MZI) switch, which is a specific instance of a MZI, comprising a pair of arms disposed between a pair of 2x2 couplers, is presented here. Within the description below embodiments of the invention are described and depicted with respect to configurations employing multimode interferometers (MMIs) as 1x2 or 2x2 optical couplers. Within other embodiments of the invention, with accordingly different derivations of electric fields, amplitudes, and transmission, a 1x2 optical coupler may employ another optical coupler element which may include, but not be limited to, an optical Y-junction, an asymmetric X-junction, a directional coupler and a zero-gap directional coupler. Within other embodiments of the invention, with accordingly different derivations of electric fields, amplitudes, and transmission, a 2x2 optical coupler may employ another optical coupler element which may include, but not be limited to, a symmetric X-junction, a directional coupler and a zero-gap directional coupler.

[0102] General Case: Electric Fields and Transmission

[0103] The schematic of the 2X2 MZI is shown in Error! Reference source not found.. The light is injected in input A, is split by the first coupler (MMI 1), travels in the two arms (wg1 and wg2), is recombined in the second coupler (MMI 2), and exits the device by outputs C and D.

[0104] The electric field (of the mode in the waveguides) will now be calculated at the different points of the device. Let ^^be the complex electric field at the entry point A of the first MMI is given by Equation (1) with ^ the electric field amplitude, ! the angular frequency, and " the time. The intensity of light at A is given by Equation (2) with # a proportionality coefficient given by Equation (3) with ^ the refractive index, $ the speed of light (in vacuum), and % thevacuum permittivity. If no light is injected in input B, then the complex electric field is givenby Equation (4).^ ()*+^ = ^ ^ (1)^

[0105] At the exit point G of the of the first MMI, the complex electric field is given by Equation (5) with / 0the coupling ratio of MMI 1, i.e. the ratio of input power coupled to the cross port, and 120the phase variation accumulated along this MMI (from A to G). At the other exit point H of the of the first MMI, the complex electric field is given by Equation (6)where the cross output of a MMI having a phase shift of − 3^ compared to the through output.^)74 = 51 − / 0 ^^ ^ 89

[0106] At the entry point J of(7) with 10the phase variation accumulated along the waveguide 1 (arm 1), from G to J. At theother entry point K of the second MMI, the complex electric field is given by Equation (8) with1^ the phase variation accumulated along the waveguide 2 (arm 2), from H to K. At the exitpoint C of the of the second MMI, there is contribution from its two entries and the complex electric field is given by Equation (9) with / ^the coupling ratio of MMI 2, and 12^the phase variation accumulated along this MMI (from J to C). At the other exit point D of the of thesecond MMI, the complex electric field is given by Equation (10).^ )7= = ^4 ^ 9 (7)

[0107] Equations (7), (8), (5), (6), and (1), these two expressions become Equations (11) and (12). The intensity of the light at C and D being given by Equations (13) and (14). Using the identity in Equation (15) leads to Equation (16) - (19) where with 1 = 1^− 10the difference of phase between the two arms.^@ = ^ ^)C(*+D789D78?EF51 − / )7051 − / ^ ^ 9 + 5 / )C705 / ^ ^ ?(3EG (11)(20) and (21), i.e. Equations (22) and (23), with the restriction given by Equation (24). ,U+VWX = @, (20)^,UYWZ[[ = B, (21)^U+VWX = 1 − / 0 − / ^ + 2 / 0 / ^ − 25 / 0 / ^C1 − / 0EC1 − / ^E cos 1 (22)

[0109] It is also possible to define an average coupling ratio and a coupling ratio difference via Equations (25) and (26) where the two coupling ratios being expressed by Equations (27) and (28). Accordingly, the cross and through transmissions can be rewritten as within Equations (29) and (30) / / 0 + / ^\]^ = (25) / ^ = / \]^ + / _ (28)UYWZ[[= 2 ` / \]^− / \^]^+ / _^(29)+ab / \^]^ − / _^cb1 − / \]^ + / _cb1 − / \]^ − / _c cos 1dU+VWX= 1 − UYWZ[[(30)

[0110] If instead of MMIs, directional couplers were employed for the 2x2 elements at either end of the MZI, the phase shift of the coupler cross output compared to the through output would be T / 2 instead of −T / 2 , but the expression of the MZI through and cross transmissions of Equations (22) and (23) would remain the same.

[0111] Switch States

[0112] When used as a switch, the cross state will occur for 1 = ^ ∗ T and the bar state for1 = C^ − 1ET, where N is an even integer. Indeed, UYWZ[[ is maximum and U+VWX minimumwhen cos 1 = 1, and U+VWXis maximum and UYWZ[[minimum when cos 1 = −1. At the cross state, the through and cross transmissions are then given by Equations (31) and (32), and forbar state these are given by Equations (33) and (34).U+VWX,Y = 1 − / 0 − / ^ + 2 / 0 / ^ − 25 / 0 / ^C1 − / 0EC1 − / ^E (31)

[0113] At the cross state, the extinction ratio expressed in dB between through and cross ports is given by Equation (35) with the value of UYWZ[[,Ydefined at Equation (32), and at the bar state, the dBextinction ratio between cross and through ports is given by Equation (36) with the value ofUYWZ[[,. defined at Equation (34).^W,Y,_^= 10 log0hUYWZ[[,Y− i(35)

[0114] The cross statenot found. for different values of / _. Interestingly, it is observed that this cross state dB extinction ratio depends strongly on / \]^, but is almost unaffected by / _. The bar state dB extinction ratio is plotted versus / _in Error! Reference source not found. for different valuesof / \]^. Interestingly, it is observed that this bar state dB extinction ratio depends strongly on / _, but is almost unaffected by / \]^.

[0115] As an example, to obtain a minimum of 30 dB extinction ratio in cross state, the average coupling ratio / \]^has to be between 0.485 and 0.515, and to obtain a minimum of 30 dB extinction ratio in bar state, the coupling ratio difference / _has to be between -0.015 and 0.015. Table 1 gives the constraints on / \]^and / _for a few extinction ratio values. Table 1: Constraints on coupler coupling ratios for several switch extinction ratios. Minimum Constraint on Constraint on average extinction ratio coupling ratio coupling ratio targeted difference 10 dB0.350 ≤ / \]^≤ 0.650 | / _|≤ 0.15020 dB0.451 ≤ / \]^≤ 0.549 | / _|≤ 0.04925 dB 0.472 ≤ / \]^≤ 0.528 | / _| ≤ 0.028 30 dB0.485 ≤ / \]^≤ 0.515| / _| ≤ 0.01535 dB0.492 ≤ / \]^≤ 0.508 | / _|≤ 0.00840 dB 0.495 ≤ / \]^≤ 0.505 | / _| ≤ 0.005 0.4985 ≤ / 50 dB\]^ / ≤ 0.0015≤ 0.5015|_|

[0116] Identical Couplers

[0117] Transmission: If the two couplers are identical, / 0= / ^, the MZI through and cross transmissions simplify as Equations (37) and (38) with / = / 0= / ^.

[0118] Extinction Ratio: In the cross state, the through and cross transmissions are given by Equations (39) and (40) whereas in the bar state they are given by Equations (41) and (42). Accordingly, in the cross state the extinction ratio, in dB, between through and cross ports is given by Equations (43) and (44) which leads to the condition defined in Equation (45) which allows expression of the coupling ratio as a function of the dB extinction ratio by Equation (46). At the bar state, the dB extinction ratio between cross and through ports is given byEquation (47).U+VWX = 1 − 2 / C1 − / EC1 + cos 1E (37)UYWZ[[ = 2 / C1 − / EC1 + cos 1E (38)U+VWX,Y = 1 − 4 / C1 − / E (39)UYWZ[[,Y = 4 / C1 − / E (40)U+VWX,. = 1 (41)UYWZ[[,. = 0 (42)

[0119] Couplers

[0120] Transmission: If the two couplers have a 50 / 50 splitting ratio, i.e. / = 0.5, the MZI through and cross transmissions simplify as given by Equations (48) and (49).

[0121] Extinction Ratio: At the cross state, the through and cross transmissions become those defined by Equations (50) and (51) and then for the bar state by Equations (52) and (53). At the cross state, the dB extinction ratio between through and cross ports is now given by Equation (54) and at the bar state, the dB extinction ratio between cross and through ports is given by Equation (55). 1− cos 1U+VWX = (48)UYWZ[[,Y = 1 (51)U+VWX,. = 1 (52)UYWZ[[,. = 0 (53)^W,Y,_^ = ∞ (54)^W,.,_^ = ∞ (55)

[0122] Waveguides with Losses

[0123] General Couplers

[0124] The case where the two arms have optical losses will now be considered. Let U0 andU^ be the transmission of respectively wg1 and wg2, the relation with their dB insertion loss,^0C_^Eand ,^^C_^Eare given by Equations (56) and (57) where U0and U^are given by (58) and (59) respectively. Now, by exploiting the study above in "Electric Fields and Transmission", the complex electric field at points J and K now become those given byEquations (60) and (61).,^0C_^E = −10 log0 U0 (56),^^C_^E = −10 log0 U^ (57)

[0125] Accordingly, at byEquations (62) and (63) whilst the complex electric field at point D is given by Equations (64) and (65) such that the intensity of the optical signals at C and D is given by Equations (66) and (67). The through and cross transmissions of the MZI become those defined by Equations (68)and (69) with the total transmission through the MZI being defined by Equation (70).^@ = 51 − / ^ 5U051 − / 0 ^ ^()*+^)789^)79^)78?(62)

[0126] Identical Couplers

[0127] If the two couplers are identical, the MZI through and cross transmissions simplify as given by Equations (71) and (72). If the two couplers have a 50 / 50 splitting, the MZI throughand cross transmissions simplify to those of Equations (73) and (74) respectively.U+VWX = U0C1 − 2 / E + CU0 + U^E / ^ − 25U0U^ / C1 − / E cos 1 (71)

[0128]

[0129] Transmission

[0130] The calculation of the transmission of a 1x2 MZI or 1x2 switch based upon a 1x2 MZI. is presented below. The 1x2 MZI comprising a pair of waveguide arms (waveguides) connected by a 1x2 coupler one end and a 2x2 coupler at the other end. With the 1x2 coupler as the input the device operates as a 1x2 MZI and with the 1x2 coupler as the output as a 2x1 MZI.

[0131] General Case : Electric Fields and Transmission

[0132] A schematic of the 1x2 MZI is depicted in Figure 4 The light (optical signal) is injected in input A, is split by the first coupler (MMI 1), travels in the two arms (wg1 and wg2), is recombined in the second coupler (MMI 2), and exits the device by outputs C and D. The calculation is similar to that above for the 2x2 MZI described within "Electric Fields and Transmission", the difference being that there is no input B and there is no phase shift between the two outputs of the first MMI (MMI 1).

[0133] At the exit point G of the of the first MMI, the complex electric field is given by Equation (75) with / 0the coupling ratio of MMI 1, i.e. now the ratio of input power coupled to the bottom port H (that can conventionally still be called "cross" port by similarity with the case of the 2X2 MZI, although this is arbitrary since the MMI 1 has only one input), 120the phase accumulated along this MMI (from A to G), and the same other quantities. At the other exit point H of the of the first MMI, the complex electric field is now given by Equation (76). The complex electric field at points J, K, C, and D are still given by Equations (7), (8), (9), and (10). Replacing the complex electric field at the different points by their values as depicted by Equations (75) and (76), the two expressions become those of Equations (77)-(80) respectively.^4=51 − / 0^^^)789(75)

[0134] that given by Equations (81) and (82). Then using Equations (20), (21), and (2), the top (port C) and bottom (port D) transmissions of the 1x2 MZI, that can conventionally still be called "through" and "cross" transmissions, are then given by Equations (83) and (84) where the condition of Equation (85) is still valid. ^@ HC 0EC T, = # ^ 1 − / 1 − / ^E + 25 / 0 / ^C1 − / 0EC1 − / ^E cos ;10 − 1^ +2<

[0135] If instead of the first MMI (1x2 MMI. MMI 1), a Y-junction is employed, the phase shift between the two outputs of this first coupler would be the same, i.e.0, and the expression of the MZI "through" and "cross" transmissions of Equations (83) and (84) would be the same.

[0136] If instead of the second MMI (2x2 MZI, MMI 2), a directional coupler was used, thephase shift of this coupler cross output compared to the through output would be 3^ instead of− 3^ . Accordingly, the complex electric field at points C and D would now be defined byEquations (86) and (87) and the MZI "through" and "cross" transmissions would now be defined by Equations (88) and (89) respectively.^@= ^ ^)C(*+D789D78?EH51 − / 051 − / ^^)79+5 / 05 / ^^);7?D3^<I(86)

[0137]

[0138] When used as a switch, the "cross" (bottom) state will occur for 1 = −^^2T^and the "bar"(top) state for 1 = 3^ ^2T^. Indeed, UYWZ[[ is maximum and U+VWX minimum when sin 1 = −1, andU+VWX is maximum and UYWZ[[ minimum when sin 1 = 1. At the "cross" state, the "through" and "cross"transmissions are then given by Equations (90) and (91) and at the "bar" state they are given by Equations (92) and (93). They are the same as for the case of the 2X2 MZI, i.e. Equations (31), (32), (33), and (34). U+VWX,Y= 1 − / 0− / ^+ 2 / 0 / ^− 25 / 0 / ^C1 − / 0EC1 − / ^E (90) UYWZ[[,Y= / 0+ / ^− 2 / 0 / ^+ 25 / 0 / ^C1 − / 0EC1 − / ^E(91)

[0139] If instead of the second MMI (MMI 2×2), a directional coupler was used, the switch "cross" state would occur for 1 =32T and the3^^ ^"bar" state for 1 = −^^2T^, but the expression of the "through" and "cross" transmissions at the "cross" "bar" states ofEquations (90), (91), (92), and (93) would remain the same.

[0140] Extinction Ratio At the "cross" state, the dB extinction ratio between "through" and "cross" ports is given by Equation (94) with the value of UYWZ[[,Ydefined at Equation (91), and at the "bar" state, the dB extinction ratio between "cross" and "through" ports is given by Equation (95) with the value of UYWZ[[,.defined at Equation (93). They are therefore the same as for the case of the 2X2 MZI. ^W,Y,_^= 10 log0 hUYWZ[[,Y−i(94)

[0141] Coupler 1×250 / 50

[0142] Transmission: If the coupler 1×2 has a 50 / 50 splitting, i.e. / 0= 0.5, the MZI "through" and "cross" transmissions simplify as Equations (96) and (97) respectively. U+VWX =12+ 5 / ^C1 − / ^E sin 1(96)

[0143] Extinction Ratio:given by Equations (98) and (99) whilst at the bar state they are given by Equations (100) and (101). Accordingly, At the "cross" state, the dB extinction ratio between "through" and "cross" ports is now given by Equations (102) and (103). This leads to Equation (104) which allows us to express the coupling ratio as a function of the dB extinction ratio in Equation (105) U+VWX,Y =12− 5 / ^C1 − / ^E(98)

[0144] Accordingly,ports is now given by Equations (106) and (107), i.e. the same as for the “cross” state. This leads to Equation (108) which allows us to express the coupling ration as a function of the dB extinction ratio by Equation (109), i.e. the same as for "cross" state.1 + 5 / C1 − / E^2 ^ ^^ )

[0145] Couplers

[0146] Transmission: If the two couplers have a 50 / 50 splitting, i.e. / 0= / ^= 0.5 , the MZI "through" and "cross" transmissions simplify as Equations (110) and (111).

[0147] Extinction Ratio: At the "cross" state, the "through" and "cross" transmissions become those defined by Equations (112) and (113) whilst at the “bar” state they are given by Equations (114) and (115). Accordingly, at the "cross" state, the dB extinction ratio between "through" and "cross" ports is now given by Equation (116) and at the "bar" state, the dB extinction ratio between "cross" and "through" ports is given by Equation (117). U+VWX =1 + sin 1(110)2U+VWX,Y= 0 (112) UYWZ[[,Y= 1 (113) U+VWX,.= 1(114)UYWZ[[,.= 0 (115) ^W,Y,_^= ∞ (116) ^W,.,_^= ∞ (117)

[0148] 3. TOLERANCE TO FABRICATION AND POLARIZATION

[0149] The process for establishing the conditions of tolerance to fabrication and polarisation is presented with respect to the width of the waveguide(s). It would be evident that the same process canbe applied to other parameters, such as the thickness of the waveguide core or waveguide refractive index for example.

[0150] Tuning by Temperature

[0151] The phases accumulated along arm 1 and arm 2 of the MZI are given by Equations (118) and (119) with ^0and ^^the lengths of wg1 and wg2, ^^^^0and ^^^^^their effective index, and ^of light (in vacuum). Thus, the difference of phase becomes that defined by Equation (120) with ^ the frequency of light. 10 = 2T^^^^0 ^0(118)^

[0152] Now, letting ^0 be^_ = ^^ − ^0 the difference of length between wg2 and wg1, ^ = ^^width of wg2, and^ = ^^ the difference of width between wg2 and wg1.

[0153] The case of a switch tuned by temperature will now be considered (heating one armwill modify the effective index which will change the phase). Let U0 be the temperature of wg1,U^ the temperature of wg2, U = U0 the temperature of wg1, and U_ = U^ − U0 the differenceof temperature between wg2 and wg1. The effective index will now be expressed as a function of waveguide width ^ and temperature U, i.e. ^^^^C^, UE. Setting an equal length for the arms, i.e. ^_= 0, Equation (120) becomes Equation (121).

[0154] Now considering at first order a linear change of effective index with temperature, the effective index can be written as Equation (122) and Equation (121) becomes Equation (123). 1=2T^^F^^^^C^ , U + U_E − ^^^^C^ − ^, U EG(121)

[0155] state. E.g.1[= 0 for a cross state, or 1[= T for a bar state (case of a 2x2 switch). From now on, the effective index will be referred to as ^^^^,^ufor TE (transverse electric) polarization and ^^^^,^^for TM (transverse magnetic) polarization, and the difference of phase betweenthe two arms as 1^ufor TE and 1^^for TM. To tune the switch for TE, i.e. setting 1^uas given by Equation (124) the temperature difference U_,^uto apply can be calculated from Equation (123) for TE, namely Equation (125). 1^u= 1[(124)switch tuned for TE, i.e. with U_= U_,^u, is then given by Equation (126). Now, let 1^be the difference between TM arm phase difference and TE arm phase difference, i.e. Equation (127). The switch would be polarization independent when 1^= 0. From Equations (126) and (124), this difference can be expressed, as a function of ^, as given by Equation (128). For this reduces to Equation (129) for ^ = 0. 2T^^^^^^^,^^ C^ZE

[0156] Subtractingwhen ^ → 0, leads to Equation (130), i.e. Equation (131). lim `1^C^E − 1^C0E^→ ^ d.

[0158] When ^^ = 0 (later referred as "stability condition"), Equation (132) leads to_7^_^C0E = 0, i.e. the value of 1^C^E can be made independent of waveguide width variationwg2 and wg1 for small values of ^. For small values of ^, the effective indexcan be expressed by Equation (133) wherein Equation (128) becomes Equation (134). Accordingly, at the stability condition (^^= 0) and for small values of ^, the polarization phase difference 1^is then given by Equation (135). ^^^^C^ − ^, U E = ^^^^C^ , U E − ^^^^^^^^ C^ , U E(133)^^^^^,^^ C^ZE

[0159] At thepolarization phase difference 1^will be independent of the difference of width ^ between wg2 and wg1 (as long as ^ remains low), making the device tolerent to fabrication variation. However, this polarization phase difference 1^will not be zero in the general case (e.g. for bar state at 1[= T), except for the balanced cross state at 1[= 0. Also, the balanced cross state (1[= 0) will not be reached without tuning, it requires the temperature difference U_,^u, as given by Equation (136), cf. Equation (125), which can also be written for small values of ^, as Equation (137).

[0160] temperature difference. Indeed, without temperature difference (i.e. U_= 0), the imbalance, i.e. the phase difference ϕ between the arms, is given by Equation (138), cf. Equation (121), which can also be written for small values of W, as Equation (139).

[0161] This imbalance is different for TE and TM is given by Equation (140) but the variation of effective index with temperature are also different for TE and TM, see Equation (141). If the ratios are the same we obtain the condition defined in Equation (142) (which is the stability condition), they compensate each other, and the balanced situation ϕ=0 for Equation (123) is reached concurrently for TE and TM. 1=2T^^F^^^^C^ , U E − ^^^^C^ − ^, U EG (138)^^^^^,^^^^ C^ , U E ^^^^^,^^ C^ZE

[0162] It should bemade equal to zero independently of 1[if_^ ¡¡,¢£_^C^ZE =_^ ¡¡,¢¤_^C^ZE. For example, a squarewaveguide would make this _^ ¡¡,¢£lead to _^ C^ , U E ≠_^ ¡¡,¢¤_^ C^ , U E, which would make the stability condition impossiblenot independent of ^.

[0163] Example 1

[0164] Using numerical simulations, and based on measured material refractive index, theeffective index ^^^^ of a waveguide is calculated versus width ^ for TE and TM at temperatureU , providing ^^^^,^uC^, U E and ^^^^,^^C^, U E, and its variation with temperature U is alsocalculated versus width, providing_^ ¡¡,¢¤_^C^E and_^ ¡¡,¢£_^C^E. The present example is based on a silicon nitride (SiN) core cladding, with a core thickness of450 nm. To obtain_^ ¡¡,¢¤_^C^E and_^ ¡¡,¢£_^C^E, a thermo-optic coefficient (refractive indexchange over _^¥¦§ (© (0 _^¥¦«?_^ = 2.5 × 10 K and _^ =1.0 × 10(© K(0. The quantities _^ ¡¡,¢¤_^ C^, U E, _^ ¡¡,¢£_^ C^, U E, and finallythencalculated numerically.

[0165] From these numerical calculations the inventors established the plots depicted in Figures 5-8 respectively, these being: • Figure 5 depicts plots of ^^^^,^uC^, U E and ^^^^,^^C^, U E; • Error! Reference source not found. depicts_^ ¡¡,¢¤ _^_^C^, U E and ¡¡,¢£_^C^, U E; • Error! Reference source not found. depicts_^ ¡¡,¢¤_^C^E and_^ ¡¡,¢£_^C^E; and • Error! Reference source not found.

[0166] A square waveguide ( ^ = 450 nm ) is polarization independent but a small difference in waveguide width between the two arms (^ ) makes the switch sensitive to polarization, and the (absolute) value of the stability coefficient ^^is quite high for this width of 450 nm, almost the highest over widths. On the other hand, as can be observed, the stability condition ^^= 0 is possible at ^ = 246 nm. It is also possible to have a sufficiently low stability coefficient ^^, thereby approaching the stability condition, by using large waveguidewidth. e.g., at ^ = 2500 nm, ^^is much lower than at ^ = 450 nm, by a ratio of 173 times. In addition, since the change of effective index with width_^ ¡¡_^is low for large width (cf. Error! Reference source not found.), the temperatureto apply to reach thebalanced cross state of a 2x2 switch (1[= 0) will then be low (ideally 0), cf. Equation (125). Using large waveguide widths, such as 2500 nm, therefore presents advantages.

[0167] Transmission Spectrum

[0168] The difference of phase between arms, with explicitly expressing the dependence of effective index on frequency, is, cf. Equation (121), given by Equation (143). At a fixed frequency ^ , it is given by Equation (144). At a fixed frequency ^ and fixed temperature U , for small values of ^ , the effective index can be expressed as Equation (145). At fixed frequency ^ and fixed waveguide width ^ , for small values of U_(or considering at first order a linear change of effective index with temperature), the effective index can be expressed as Equation (146). Accordingly, the phase difference, Equation (144), becomes Equation (147). 1=2T^^F^^^^C^, ^ , U + U_E − ^^^^C^, ^ − ^, U EG(143)

[0169] transmission spectrum, but this would require the knowledge of­^ ¡¡ ­^ ¡¡­^and­^for all the different frequencies ^ . Instead, the following calculations

[0170] The Taylor series expansion to second order of the effective index at frequency ^, waveguide width ^, and fixed temperature U , is given by Equation (148) with ^ a reference frequency and ^ = ^ − ^ . Second order term­?^ ¡¡­^?will be neglected, i.e. the expression is ? valid for small values of ^ compared to ^ , or­ ^ ¡¡dispersion. Second order term­^?will also be neglected, i.e. the expression is valid for small values of ^ compared to ^ . The expression of Equation (148) then becomes Equation (149).^^^^C^, ^, U E = ^^^^C^ + ^, ^ − ^, U Edispersion can then be expressed as Equation (151). Using Equation (151), Equation (149) becomes Equation (152) which reduces as given by Equations (153) and (154). ^^C^, ^, U ^^^¬^E= ^C^, ^, UE+ ^^^^¬^C^, ^, UE (150))temperature U, and fixed waveguide width ^ , is given by Equation (155). Second order term­?^ ¡¡­^? will be neglected, i.e. the expression is valid for small values of ^ compared to ^ , or fordispersion. Second order term­?^ ¡¡­^?will also be neglected, i.e. the expression is valid for small values of U_compared to U , linear change of effective index with temperature.The expression of Equation (155) becomes Equation (156). Using Equation (151), Equation(156) becomes Equation (157) and therein Equation (158).^^^^C^, ^ , UE = ^^^^C^ + ^, ^ , U + U_E^^^^C^= ^^^^C^ , ^ , U E + ^¬^^^^¬^C^ , ^ , U E + U_¬^^^^C^ , ^ , U E ^^^^C^ + ^, ^= ^^^^C^ , ^ , U E +^^ F^^C^ , ^ , U E − ^^^^C^ , ^ , U EG= °1 −^^ ± ^^^^C^ , ^ , U E +^^ ^^C^ , ^ , U E

[0172] Thewhich using Equations (158) and (154) becomes Equations (160) and (161) respectively. 1=2T^C^ + ^EF^^^^C^ + ^, ^ , U + U_E − ^^^^C^ + ^, ^ − ^, U EG (159)1 =2T^^²°2 −^^± s^¬^^^^¬^ C^ , ^ , U E + U_¬^^^^C^ , ^ , U Et

[0173] difference of phase between arms (1) can be^ using Equation (161), replacing ^^^^by ^^^^,^uor ^^^^,^^, and ^^by ^^,^uor ^^,^^, for TE or TM polarizations, respectively. The switch transmission spectra can then be calculated using Equations (22) and (23) for the case of a 2x2 switch, and using Equations (83) and (84) for the case of a 1x2 switch.

[0174] Example 2

[0175] Based on the same materials as the example above within "Error! Reference source not found.", the through and cross transmission spectra (in TE and TM) of a 2x2 switch tuned at the bar state is plotted in Figures 9A-9B, Figures 10A-10B, Figures 11A-11B and Figures 12A-12B for waveguide widths ^ of 246, 450, 700, and 2500 nm, respectively, and with awaveguide width difference ^ of 0 nm and 2.0 nm (the length of wg1 and wg2 is ^ =400 µm). Figures 9A, 10A, 11A and 12A being those for the waveguide difference ^ of 0 nmand Figures 9B, 10B, 11B and 12B being those for the waveguide difference ^ of 2 nm.

[0176] The switch is tuned by adjusting the temperature difference U_to get the minimum cross transmission (TE / TM compromise) at ^ = 1311 nm . If U_> 0 the arm 2 (wg2) is heated, and if U_< 0 the arm 2 (wg2) is cooled or the arm 1 (wg1) is heated.

[0177] The extinction ratio is in the order of 17, 14, 17, and 27 dB, for the waveguide width of 246, 450, 700, and 2500 nm, respectively (with a waveguide width difference of 2.0 nm). For the waveguide width of 450 and 700 nm, the results vary when the waveguide widthdifference ^ varies, as can be seen from the different extinction ratio between ^ = 0 and^ = 2 nm (cf. Figures 10A and 10B and Figures 11A and 11B). But for the waveguide widthof 246 and 2500 nm, the extinction ratio remains the same for ^ = 0 and ^ = 2 nm (cf. Figures 9A-9B and Figures 12A-12B). For these two waveguide widths the stability condition (^^= 0) was reached or almost reached, making them tolerent to fabrication variation, i.e. not sensitive to waveguide width difference ^, as described in " Error! Reference source not found.".

[0178] For the cross state the spectra for the waveguide widths (246, 450, 700, and 2500 nm) when the waveguide width difference ^ is zero, are depicted in Figures 13A, 14A, 15A and16A respectively, because the target phase difference 1[is zero. For ^ = 2 nm , the waveguide widths 246 and 2500 nm still have a high extinction ratio, cf. Figures 13B and 16B respectively, compared to 450 and 700 nm, cf. Figures 14B and 15B respectively, again being less sensitive to waveguide width difference ^.

[0179] Polarisation Dependent Loss (PDL)

[0180] The polarization dependent loss (PDL) in through and cross ports is given by Equations (162) and (163) where the transmissions for each port and polarisation are given by Equations (164)-(167) with U+VWX,^uand U+VWX,^^detailed at Equation (22) (case of a 2x2 switch) or Equation (83) (case of a 1x2 switch), using 1 = 1^u(for TE) or 1 = 1^^(for TM), cf. Equation (161), and with UYWZ[[,^uand UYWZ[[,^^detailed at Equation (23) (case of a 2x2 switch) or Equation (84) (case of a 1x2 switch). ^^^+VWX,_^= KU+VWX,^^,_^− U+VWX,^u,_^K (162)

[0181] Extinction

[0182] The dB extinction ratio between through and cross ports is given by Equation (168) and can be calculated using Equations (22) and (23) for the case of a 2x2 switch and Equations (83) and (84) for the case of a 1x2 switch, and using Equation (161). It can therefore be expressed as a function of frequency ^ and / or temperature difference U_. This extinction ratio, as defined, is highest (positive value, e.g.30 dB) at cross state, and lowest (negative value, e.g. -30 dB) at bar state. A high absolute value is desirable (for use as a switch state). ^W,_^ = 10 log0 °UYWZ[[±, (168)

[0183] Example 3

[0184] The extinction ratio of a 2x2 switch is plotted versus temperature difference U_in Figures 17A, 18A, 19A, and 20A for waveguide widths ^ of 246, 450, 700, and 2500 nm, respectively, with a waveguide width difference ^ of 0 nm. The extinction ratio of a 2x2 switch is plotted versus temperature difference U_in Figures 17B, 18B, 19B, and 20B for waveguide widths ^ of 246, 450, 700, and 2500 nm, respectively, with a waveguide width difference ^ of and 2.0 nm. These being shown for TE and TM, and at two differentwavelengths (1291 and 1311 nm). The extinction ratio varies periodically with U_, with high positive values and high negative values corresponding respectively to cross and bar states.

[0185] When the waveguide width difference ^ is zero, the extinction ratio is high (infinite) at U_= 0 (cross state by default, without tuning) for all waveguide widths (246, 450, 700, and 2500 nm) cf. Figures 17A, 18A, 19A and 20A, because the phase difference 1 is then zero, cf.Equation (161). This being true for TE and TM, and for the different frequencies. Still at ^ =0, the extinction ratios of the bar states are not overlapping for TE and TM because ­^ ¡¡­^ ispolarization dependent in Equation (161), except for ^ = 450 nm (square waveguide). They are not overlapping for the different frequencies either, cf. Equation (161).

[0186] For ^ = 2 nm, for the waveguide widths 450 and 700 nm, the extinction ratios are similarly non-overlapping for TE and TM, even for the balanced cross state (1 = 0), cf. Figures 18B and 19B respectively. However, for the waveguide widths 246 and 2500 nm, the extinction ratios are still overlapping for TE and TM for the balanced cross state, cf. Figures 17B and 20B respectively. For these two waveguide widths the stability condition (^^= 0) was reached or almost reached, making them tolerent to fabrication variation, i.e. not sensitive to waveguide width difference ^, as described in "Tolerance to Fabrication and Polarization".

[0187] For ^ = 246 nm, the balanced cross state (1 = 0) does not occur at U_= 0, cf. Figure 17B, because the phase imbalance due to the difference of width ^ needs to be compensated by applying a temperature difference, cf. "Tolerance to Fabrication and Polarization". For ^ = 2500 nm, the balanced cross state still almost happen at U_= 0, cf. Figure 20B, because the value of­^ ¡¡­^is much lower at ^ = 2500 nm than ^ = 246 nm, cf.Error! Reference source not found., in Equation (137). For ^ = 246 nm and ^ =2500 nm, the extinction ratios of the bar states are not overlapping for TE and TM but theirdifference remains the same when the waveguide width difference ^ varies, whereas for ^ =450 nm and ^ = 700 nm their difference also varies.

[0188] Free Spectral Range

[0189] Expression of FSR: Considering the difference of phase between arms of Equation (161), the difference of temperature U_,[to apply to obtain a target phase difference 1[(for a specific switch state, e.g.1[= T) at frequency ^ is given by Equation (169). In order to obtain it at the reference frequency ^ = ^ , the difference of temperature required is given by Equation (170). The difference of phase between arms at this temperature difference U_,[C^ E, i.e. for the switch tuned at a specific state (of phase difference 1[) for the frequency ^ , is, using Equation(170) in Equation (161) to yield Equation (171). Differentiating this phase 1 with respect to frequency ^ gives Equation (172) which reduces to Equation (173) noting that­^ ¡¡­^C^ , ^ , U E,­^®­^C^ , ^ , U E,­^ ¡¡­^C^ , ^ , U E, and­^®­^C^ , ^ , U E are not dependent onzero.$1[ − ^ s°2 − ^^ ± ¬^^^^¬^ C^ , ^ , U E + ° ^^ − 1± ¬^^ C^ , ^ , U Edifferences·7·^and setting the difference of phase ¸1 = 2T, the difference of frequency ¸^ is then the spectral range (FSR). Thus, the FSR, for the switch tuned for the frequency ^ andfor the parameters expressed at ^ is given by Equation (175). In general, the highest possible FSR is sought, providing higher bandwidth for the switch. The FSR is infinite if the denominator in Equation (175) is zero. This leads to Equation (176) or Equation (177). ¬^^2T^^ ¬^^^^ C^ , ^ , U Einfinite if a waveguide width difference is present (^ ≠ 0), cf. Equation (175). For the general case, the FSR can be infinite if the specific condition on ^ of Equation (177) is satisfied. But with the fabrication variation, ^ is uncontrolled, so it would happen only by chance. For a given value of ^, it would happen at a specific phase 1[given by Equation (176), which in this case will not correspond to a specific switch state (or only by chance).

[0192] When the FSR is infinite, the extinction ratio will be the same for all the frequencies. In the plots of extinction ratio versus temperature difference, theoretically described in “Error! Reference source not found." at Equation (168), or experimentally obtained (in this case possibly versus power applied to a heater), the point where different curves for different frequencies cross will thus indicate the situation of infinite FSR. From the periodicity of these curves, the phase 1[at which it happens can be extracted, and it is then possible to evaluate the experimental waveguide width difference with Equation (177).

[0193] It would also be possible to get an infinite FSR for a specific switch state (at phase1[), or between two switch states to get the maximum bandwidth for both, by intentionallyapplying a waveguide width difference ^, but this difference would have to be higher enough than the one from fabrication variation. In a similar way to the stability coefficient ^^of Equation (132), another stability coefficient ^^,¯can be defined by Equation (178). As seen from the term multiplying ^ in the denominator of Equation (175), the FSR could be made independent of waveguide width difference ^ if ^^,¯= 0. ^^,¯ =¬^^^^C^ , ^ , U E¬^^ ¬^^^^ ¬^^¬^ C^ , ^ , U E −¬^ C^ , ^ , U E C^ , ^ , U E(178)

[0195] The FSR of a switch is displayed in Figures 21A to 21D versus waveguide width difference ^ for waveguide widths 246 nm, 450 nm, 700 nm and 2500 nm respectively. It isplotted for different target phase differences 1[ (corresponding to specific switch states, e.g.1[ = T for the bar state of a 2x2 switch). As seen on the figure, and as expressed in Equation(175), it varies with ^. For 1[= 0 (cross state of a 2x2 switch), the FSR is infinite at ^ = 0. In other cases, the FSR increases or decreases when ^ differs from 0. So, since the sign of ^ is uncontrolled with fabrication variation, the FSR will reduce when a waveguide width difference is present, with one chance over two.

[0196] Referring to Figure 22 there is depicted a plot of ^^,¯versus waveguide width ^. Ascan be seen, the stability condition ^^,¯ = 0 is possible at ^ = 449 nm for TE and ^ =440 nm for TM, so almost at the same width. At the nearby width of 450 nm, as can be seenin Figure 21B, the FSR is almost constant versus waveguide width difference ^. It is also possible to have a low enough stability coefficient ^^,¯, thereby approaching the stability condition, by using large waveguide width, such as ^ = 2500 nm. The FSR does not vary much with ^, cf. Figure 21D. At ^ = 0, or near it, the FSR is also higher for larger waveguide widths, as seen in Figures 21A-21D for the different waveguide widths. This is due to the effect ¼½®of the ratio¼½¼¢ ¡¡decreasing with waveguide width (for ^ > 300 nm), cf. Error! Reference ¼¢source in Equation (175). As will become evident large waveguide widths, such as 2500 nm, present additional advantages.

[0197] Comparison of Switches

[0198] The case of equal waveguide widths (^ = 0) will now be considered. Under this condition FSR is then given by Equation (179). For the 2x2 switch, at the balanced cross state (1[= 0) the FSR is given by Equation (180) and at the lowest unbalanced bar state (1[= ±T) by Equation (181).

[0199] ¬^^^^ C^ , ^ , U Ebandwidth over which the extinction ratio is above a given level is directly proportional to the FSR. So, to maximize the bandwidth, the balanced cross and the lowest unbalanced bar state should be used, and higher unbalanced states should be avoided as they would reduce the bandwidth. For the 1x2 switch, at the lowest unbalanced "cross" state (1[= −3^) and at the lowest unbalanced "bar" state (1[=3^) the FSR is given by Equation (182).

[0201] At higher unbalanced cross and bar states, the FSR will also get lower. As can be seen, the FSR of a 1x2 switch (^^º3 / ^) is twice the FSR of a 2x2 switch (^^º3, worst of cross and bar states). In the same way, the bandwidth of a 1x2 switch is twice the one of a 2x2 switch. Consequently, for a given frequency within this bandwidth, the extinction ratio of a 1x2 switch will be higher than the one of a 2x2 switch.

[0202] Polarization Independence

[0203] Conditions: A switch where each arm has two sections will now be considered, a first section ^ of length ^^with the width ^ = ^^, and a second section ^ of length ^^with the width ^ = ^^. The phases accumulated along arm 1 and arm 2 is now defined by Equations (183) and (184) with ^^^^0^, ^^^^0^, ^^^^^^, and ^^^^^^respectively the effective index of wg1 in section ^ and section ^ and wg2 in section ^ and section ^. With the same considerations as in the previous section, the difference of phase is now given by Equation (185) which with the appropriate substitutions becomes Equations (186) and (187) respectively. For small values of ^, this becomes Equations (188) and (189). Whilst a common phase shifter element is considered within the analysis for simplicity it would be evident that different phase shifters may be applied to section A and section B such that independent control is provided allowing finer compensation.10 =2T^b^^^^0^ ^^ + ^^^^0^ ^^c(183)temperature difference U_,^uto apply can be calculated from Equation (189) for TE, as given by Equation (191). Accordingly, The phase difference for TM, calculated using Equation (189) for TM, with the switch tuned for TE, i.e. with U_= U_,^u, is then given by Equation (192). As a result the polarization phase difference 1^, Equation (127), is now written as Equation (193). 1^u= 1[(190)1 2T^^^ = ^^ ^ h$ Às^^^^^^^,^^^^ C^^ , U E^^^,^u^ ^^^^^,^uindependent, i.e. 1^= 0 , for any state, i.e. independently of the value of 1[, if the two conditions defined in Equations (194) and (195) are satisfied. The first condition, Equation (194), leads to Equations (196) and (197) with respect to the lengths of the two waveguide sections. The second condition, Equation (195), using Equation (196) leads to Equation (198) which when combined with Equation (197) leads to Equation (199). s^^^^^^^,^^C^^E + ^^^^^^^,^^C^ Etthe waveguide widths ^^and ^^that satisfy the condition ^^,^^C^^, ^^E=0 would make theswitch polarization independent, with the ratio of lengths of section B and section A determined by Equation (197). ^^,^^C^^, ^^E = s^^^^^,^uC^^E −^^^^^,^^C^^Et s^^^^^,^^^^C^^, U E

[0208] Figure 23 depicts the stability coefficient ^^,^^ versus the waveguide widths ^^ and^^. As can be observed, a line of widths (^^, ^^) is present on the graph, for which ^^,^^=0. Figure 24 depicts the length ratio ~y~Â versus the waveguide widths ^^ and ^^. Some regions(in white) correspond to aand are therefore not physical solutions. A cut of the two-dimensional (2D) map for ^^,^^and~y~Âare presented in Error! Reference source not found.A and 25B respectively fornm.

[0209] At this waveguide width of 2500 nm for section ^, the stability condition ^^,^^= 0is possible at ^ = 275 nm, cf. Figure 25A. The corresponding length ratio is ~y^ then ~Â =0.553, cf. Figure 25B. For a length ^ = ^^ + ^^ of 400 µm, this corresponds to ^^ = 258 µmand ^^= 142 µm.

[0210] In practice, the waveguide width difference ^ due to fabrication variation may be different for section ^ and section ^. But choosing a waveguide width ^^= 2500 nm for section ^ the inventors are already making the MZI design tolerant to fabrication variation as seen in "Tolerance to fabrication and polarization". Section ^ is used for compensating thedifference of variation of effective index with temperature _^ ¡¡_^ between TE and TM of section^, at the origin of polarization phase difference 1^, cf.(135) and (194), by providingan opposite one.

[0211] Transmission Spectrum

[0212] The transmission spectrum will now be calculated, in the same way as done above in earlier section “Transmission Spectrum”. The difference of phase between arms, with explicitly expressing the dependence of effective index on frequency, is, cf. Equation (186), given by Equation (201). At a fixed frequency ^ , fixed temperature U , and fixed waveguide widths ^^and ^^, for small values of ^ and for small values of U_(or considering at first order a linear change of effective index with temperature), the difference of phase between arms is, cf. Equation (189), given by Equation (202). 2T^1 = ¾F^^^^C^, ^^, U + U_E − ^^^^C^, ^^ − ^, U EG^^

[0213] Equation (203) and using the same considerations as in “Transmission Spectrum” above, cf.Equations (158) and (154), this becomes Equation (204) which becomes Equation (205).1= ^3C^^D¯E Y¾F^^^^C^ + ^, ^^, U + U_E − ^^^^C^ + ^, ^^ − ^, U EG^^ +phase between arms ( 1 ) can^ using Equation (205), replacing ^^^^by ^^^^,^uor ^^^^,^^, and ^^by ^^,^uor ^^,^^, for TE or TM polarizations,respectively. The switch transmission spectra can then be calculated using Equations (22) and (23) for the case of a 2x2 switch, and using Equations (83) and (84) for the case of a 1x2 switch.

[0215] Example 6

[0216] Using the parameters determined in “Condition” above to obtain polarization insensitivity, ^. ^. ^^= 2500 nm , ^^= 275 nm , ^^= 258 µm , and ^^= 142 µm , the through and cross transmission spectra (in TE and TM) of a 2x2 switch tuned at the cross andbar states are depicted in Figures 26A and 26B respectively for a waveguide width difference^ = 2 nm and with realistic non-ideal MMI 2×2 couplers ( / 0 = 0.51, / ^ = 0.52).

[0217] As evident, TE and TM spectra are now aligned, even when ^ ≠ 0 and 1[≠ 0, showing fabrication tolerance and polarization insensitivity. The worst-case extinction ratio over the L2 band (1304.5-1317.5 nm) is 30 dB for cross state and 34 dB for bar state, showing improvement from the values of 30 dB and 25 dB, respectively, when only one section is used (^ = 2500 nm), cf. Figures 37A and 37B respectively for the cross and bar states.

[0218] In order to establish the sensitivity to fabrication precision, the spectra were also calculated for ±10 nm of the waveguide width of section ^, i.e. ^^= 265 nm (Figures 27A and 27B) and ^^= 285 nm (Figures 28A and 28B). These result in the same extinction ratio of 30 dB for cross state and a drop to 31 dB for bar state, which is still better than when only one section is used.

[0219] Extinction Ratio

[0220] The dB extinction ratio between through and cross ports is given by Equation (206) and can be calculated using Equations (22) and (23) for the case of a 2x2 switch and Equations (83) and (84) for the case of a 1x2 switch, and using Equation (205). It can therefore be expressed as a function of frequency ^ and / or temperature difference U_. This extinction ratio, as defined, is highest (positive value, e.g.30 dB) at cross state, and lowest (negative value, e.g. -30 dB) at bar state. A high absolute value is desired (for use at a switch state). ^W,_^ = 10 log0 °UYWZ[[±, (206)

[0221] Example 7

[0222] The extinction ratio of a 2x2 switch is plotted versus temperature difference U_inError! Reference source not found. for ^^ = 2500 nm , ^^ = 275 nm , ^^ = 258 µm ,^^ = 142 µm, a waveguide width difference ^ = 2 nm, and with realistic non-ideal MMI2×2 couplers ( / 0= 0.51, / ^= 0.52). The extinction ratio curves of TE and TM are now overlapping, even when ^ ≠ 0 and 1[≠ 0, showing fabrication tolerance and polarization insensitivity.

[0223] Tuning by Arm Length

[0224] Tolerance to Fabrication and Polarization

[0225] The case of a switch tuned by arm length will be considered in this section (changing the length of one arm will change the phase). A length increase could be provided by pulling a suspended waveguide with a MEMS for example. With the same notations as in “Tolerance to Fabrication and Polarization” then Equation (120) becomes Equation (207). For small values of ^, the effective index can be expressed as Equation (208) wherein Equation (207) becomes Equation (209). Accordingly, To tune the switch for TE, i.e. setting 1^u= 1[, the length difference ^_,^uto apply can be calculated from Equation (209) for TE, as given by Equation (210). 1=2T^FC^ + ^_E^^^^C^ E − ^^^^^C^ − ^EG.(207)

[0226] Accordingly, the phase difference for TM, calculated using Equation (209) for TM, with the switch tuned for TE, i.e. with ^_= ^_,^u, is then given by Equation (211). The polarization phase difference 1^= 1^^− 1^uis now defined by Equation (212).

[0227] The inventors define a stability coefficient ^^,~as given by Equation (213) wherein when ^^,~= 0 (stability condition), the polarization phase difference, Equation (212),becomes Equation (214).^3^~^ _^ ^ C^^E _^ ^ C^^E1^^ =Ys  ¡¡,¢£_^C^ E −  ¡¡,¢£  ¡¡,¢¤^ ¡¡,¢¤C^^E _^C^ Et +  ¡¡,¢£^ ¡¡,¢¤C^^E1[. (211))polarization phase difference 1^will be independent of the difference of width ^ betweenwg2 and wg1 (as long as ^ remains low), making the device tolerant to fabrication variation. However, this polarization phase difference 1^will not be zero in the general case (e.g. for bar state at 1[= T), except for the balanced cross state at 1[= 0. Also, the balanced cross state (1[= 0) will not be reached without tuning, it requires the length difference ^_,^u, cf. Equation (210), as given by Equation (215). The imbalance due to the difference of width ^ is canceled by applying this length difference. Indeed, without length difference (i.e. ^_= 0 ), the imbalance, i.e. the phase difference 1 between the arms, is, cf. Equation (209), then given by Equation (216). This imbalance is different for TE and TM, as given by Equation (217), but the effective index is also different for TE and TM, as given by Equation (218) and if the ratios are the same, c.f. Equation (219) (which is the stability condition), they compensate each other, and the balanced situation 1 = 0 for Equation (209) is reached concurrently for TE and TM. −^^ ^^^^^,^u^^C^ E

[0229] The polarizationindependently of 1[if ^^^^,^^C^ E = ^^^^,^uC^ E . A square waveguide would make this possible, but thisC^ E, which would make the stability condition impossible to obtainof ^.

[0230] Example 8

[0231] The stability coefficient ^^,~is plotted versus waveguide width ^ in Error! Reference source not found.. A square waveguide (^ = 450 nm) is polarization independent but a small difference in waveguide width between the two arms (^ ) makes the switch sensitive to polarization, and the (absolute) value of the stability coefficient ^^,~is quite high for this width of 450 nm, almost the highest over widths. On the other hand, as can be observed, the stability condition ^^,~= 0 is possible at ^ = 273 nm . It is also possible to have a sufficiently low stability coefficient ^^,~, thereby approaching the stability condition, by usinglarge waveguide width. E.g., at ^ = 2500 nm, ^^,~is much lower than at ^ = 450 nm, by a ratio of 160 times. In addition, since the change of effective index with width_^ ¡¡_^is low for large width (cf. Error! Reference source not found.), the length difference to apply to reach the balanced cross state of a 2x2 switch (1[= 0) will then be low (ideally 0), cf. Equation (210). Using large waveguide widths, such as 2500 nm, therefore presents advantages.

[0232] Transmission spectrum, PDL, extinction ratio, and FSR can be calculated in a similar manner as above in “Transmission Spectrum”. “PDL”, “Extinction Ratio” and “FSR.” Similarly, the polarization insensitivity conditions can be calculated in a similar manner as above in “Polarisation Insensitivity.”

[0233] 4.2X2 QUAD-SWITCH

[0234] Properties

[0235] Description and States

[0236] A 2x2 switch can be formed using four 1×2, sub-switches as illustrated in Figure 31. It is referred as a “2x2 quad-switch” by the inventors. For the cross state, the sub-switches 1×2 A and C are set on "bottom" state ("cross" state) and the sub-switches 1×2 B and D on "top" state (that will arbitrarily be taken as "cross" state here), thereby sending the light from port A to port D (through ports N and U) and the light from port B to port C (through ports P and S), cf. Figure 32. For the bar state, the sub-switches 1×2 A and C are set on "top" state ("bar" state) and the sub-switches 1×2 B and D on "bottom" state (that will arbitrarily be taken as "bar" state here), thereby sending the light from port A to port C (through ports M and R) and the light from port B to port D (through ports Q and V), cf. Figure 33.

[0237] Extinction Ratio

[0238] When set to a defined switch state, at the output of the first 1x2 sub-switch (A or B) the light will mainly be in one waveguide only (with the extinction ratio between ports of the 1x2 switch), and at the input of the second 1x2 sub-switch (C or D) the light will mainly be collected from one waveguide only (with the extinction ratio between ports of the 1x2 switch). So, the extinction ratio between ports of the 2x2 quad-switch will be the double of the one of the 1x2 switch.

[0239] More precisely, with ,^, ,^, ,@, ,B, ,^, ,Ã, ,Ä, ,Å, ,Æ, ,Ç, ,È, and ,É, the intensity of light at ports A, B, C, D, M, N, P, Q, R, S, U, and V, respectively, and with U+VWXand UYWZ[[the "through" and "cross" transmissions of the 1x2 sub-switch, cf. Equations (83) and (84), the intensities of light after going through the first sub-switches 1×2 at ports M, N, P and Q are given by Equations (220)-(223) respectively and the intensities at the output of the 2x2 quad-switch at ports C and D by Equations (224) and (225). These can re-written therefore as Equations (226) and (227). ,^= ,Æ= U+VWX,^, (220) ,Ã= ,È= UYWZ[[,^, (221) ,Ä= ,Ç= UYWZ[[,^, (222) ,Å= ,É= U+VWX,^, (223) ,@= U+VWX,Æ+ UYWZ[[,Ç, (224) ,B= UYWZ[[,È+ U+VWX,É, (225)

[0240] Looking at thethrough and cross transmissions of the 2x2 quad-switch are given by Equations (228)-(231) respectively. Equivalently, with the light injected in input B (,^= 0), the through and cross transmissions of the 2x2 quad-switch are the same, Equations (229) and (231). The dB extinction ratio (defined here as positive or negative value) between the output through and cross ports is given by Equation (232) which , using Equations (229) and (231) gives Equation (233) with ^W,_^the extinction ratio of the 1x2 switch, cf. Equation (168). U+VWX,ÊX\_ =,@, (228)^

[0241] Looking at the output C, with the light injected in inputs A and B, the ratio of power coming from A and B is, cf. Equation (226), given by Equation (234), so, the dB extinction ratio (or crosstalk) between the input through and cross ports is, with equal power ,^= ,^, given by Equations (235) which becomes Equation (236). ^=U ^YWZ[[ ,^,(234) U^+VWX ,^^′W,ÊX\_,_^ = 10 log0 hU ^YWZ[[^i,(235)

[0242] Equivalently, looking at the output D, with the light injected in inputs A and B, the dB extinction ratio between the input through and cross ports is the same, Equation (236).

[0243] The 2x2 quad-switch extinction ratio between ports, looking at output ports or input ports, is the same, which is expected since the device is symmetrical. This extinction ratio is the double of the one of the 1x2 switch.

[0244] Comparison of 2x2 quad-switch with standard 2x2 Switch

[0245] Since the extinction ratio of a 1x2 switch is higher than the one of a 2x2 switch (cf. “Comparison of Switches”), the extinction ratio of the 2x2 quad-switch is more than the double of the one of a 2x2 switch. The extinction ratio of the 2x2 quad-switch being the double of the one of the 1x2 switch, its bandwidth will be larger. And since the bandwidth of a 1x2 switch is twice the one of a 2x2 switch (cf. "

[0246] Comparison of Switches"), the bandwidth of the 2x2 quad-switch will be more than twice the one of a 2x2 switch. These are two main advantages of the 2x2 quad-switch compared to the (standard) 2x2 switch.

[0247] Another point is that the phase difference 1[to reach a cross or bar state (worst-case) is |1[| =3^ for the 2x2 quad-switch (same as for the 1x2 switch) and |1[| = T for the 2x2 switch. The polarization phase difference 1^, Equation (127), will consequently be lower for the 2x2 quad-switch than for the 2x2 switch, cf. Equation (128). At the stability condition (^^= 0), the polarization phase difference 1^of the 2x2 quad-switch will be half the one of the 2x2 switch, cf. Equation (135), making it therefore more tolerant to polarization.

[0248] In addition, for the same reason that the phase difference 1[to apply to tune the 2x2 quad-switch is half the one of the 2x2 switch, the temperature difference to apply to tune the switch will be lower for the 2x2 quad-switch than the 2x2 switch, cf. Equation (125), and will be half if the difference of effective index between the two arms is negligible, cf. Equation (125) or (169). This is advantageous as it makes the materials used less susceptible to degradation, or, for a given temperature, it allows for a reduced length (^) of the waveguide, which in turn also reduces the polarization phase difference 1^when the stability condition (^^= 0) cannot be completely reached, cf. Equation (134), thus increasing the polarization tolerance, and possibly reduces the average waveguide width difference ^ due to fabrication variation, which will additionally reduce the polarization phase difference 1^again, cf. sameEquation (134). It would be evident that the lower the temperature difference to apply implies lower electrical power and lower electrical current which is also advantageous as it helps prevent micro-heater degradation from high current densities.

[0249] Another benefit of exploiting wide waveguide sections is improved reproducibility in the temperature difference required to reach a certain switch state for the case of thermo- optic effect, and therefore of the electrical power of the micro-heater(s) if employed. One consequence of this is the ability to be at a predefined state by default (without any applied phase tuning) and therefore without power consumption. For example, the 2x2 switch can be at cross state by default, and the 1x2 switch can be at a state of equal splitting by default. Another consequence is the possibility to group phase shifters (sharing the same control) for the 2x2 quad-switch, such as one group of four for cross state and another group of four for bar state, since the four 1x2 sub-switches are switched together.

[0250] It is also possible, as depicted in Figure 47, to establish a 2x2 quad-switch employing 4 1x2 sub-switches to eliminate waveguide cross-overs within the 2x2 quad-switch. As depicted four 1x2 sub-switches 4700A to 4700D are coupled in sequence to input ports A, D, B and C respectively. The ports M and N of first 1x2 sub-switch 4700A are coupled to fourth and second 1x2 sub-switches 4700D and 4700B respectively at ports R and U whilst the outputs P and Q of the third 1x2 sub-switch 4700C are coupled to ports S and V respectively of the fourth and second 1x2 sub-switches 4700D and 4700B respectively. The waveguide routings for ports M-R, N-U, V-Q and P-S do not cross.

[0251] Within the 2x2 quad-switch it is possible to establish other switch states such as an off- state where no light exists the device. This can be obtained, with reference to Figure 31, by setting the 1x2 sub-switch A to what the inventors refer to as a "bottom" state, thereby sending the light from port A to port N, setting the 1x2 sub-switch B to what the inventors refer to as a "top" state, thereby sending the light from port B to port P. If now we set the 1x2 sub-switch C to the "top" state, thereby collecting the light from port R (where no light is present from 1x2 sub-switch A) to port C, and setting the 1x2 sub-switch D to the "bottom" state, thereby collecting the light from port V (where no light is present from the 1x2 sub-switch B) to port D.

[0252] There is also the possibility to establish the 2x2 quad-switch into a mixed state, such as a cross or bar state for one input and off state for the other input. For example, by setting the 1x2 sub-switch A to the "bottom" state, 1x2 sub-switch B to the "top" state, 1x2 sub-switch C to the "top" state, and 1x2 sub-switch D to the "top" state, the 2x2 quad-switch will be in barstate for input A and off state for input B. Other configurations would be evident to one of skill in the art.

[0253] It would also be evident that the 2x2 quad-switch provides for variable attenuation and signal mixing by adjusting progressively the phase of each 1x2 sub-switch.

[0254] The inventors also note that by appropriate design that unequal tuning may be applied to the four 1x2 sub-switches as it is evident from the analysis presented within this specification that applying a slightly different tuning to some of the 1x2 sub-switches, for example on the order of few percent, will have the effect of establishing a pair of low transmission dips slightly offset in wavelength such that the initial single low transmission dip is split spectrally into two dips which has the effect of increasing the operating bandwidth of the quad 2x2 switch. This also offers stability to the 2x2 quad-switch against fabrication variation (e.g. width of electrical micro-heater, suspension of waveguide for thermal insulation etc.), particularly in the instance where the electrical micro-heaters may be grouped, as slightly different tuning would naturally occur from inherent manufacturing variations within the 1x2 sub-switches.

[0255] Additionally, slightly different tuning can also be used on purpose to increase the performance of the 2x2 quad-switch. This can be implemented for example by a slight modification of the width of electrical micro-heater (resulting in a slightly different resistance, power, and temperature), while still having the micro-heaters grouped electrically for control (e.g. in parallel or in series).

[0256] Example 9 and Comparison

[0257] Transmission Spectrum and PDL

[0258] The through and cross transmission spectra (in TE and TM) of a (standard) 2x2 switch (waveguide width of 2500 nm) tuned at the cross and bar states is shown in Figures 34A and 34B respectively for a waveguide width difference ^ = 0 (no fabrication variation) andwith ideal MMI 2×2 couplers ( / 0 = / ^ = 0.5). The cross and bar state transmission spectra for^ = 2 nm (to represent fabrication variation) are presented in Figures 35A and 35Brespectively. Figures 36A and 36B respectively depict the transmission spectra with non-ideal MMI 2×2 couplers ( / 0= 0.51, / ^= 0.52) whilst Figures 37A and 37B depict the spectra for both non-idealand waveguide width difference ^ = 2 nm. The average couplingratio of the non-ideal MMI 2×2 couplers is then / \]^ = 0.515 and the coupling ratio difference / _ = 0.005, cf. Equations (25) and (26). Figures 37C and 37D depict zoomed views of themain output depicted in Figures 37A and 37B.

[0259] The worst-case extinction ratio over the L2band (1304.5-1317.5 nm) is infinite at cross state and is 25 dB at bar state for ^ = 0 and ideal couplers (Figures 34A and 34B). When a waveguide width difference ^ = 2 nm is introduced, the extinction ratio is now 58 dB at cross state, which is still high, and stays at 25 dB at bar state (Figures 35A and 35B) due to the use of large waveguide width (2500 nm). With realistic couplers, the extinction ratio drops to 30 dB at cross state. It remains at 25 dB at bar state, but the curves get smoother (the maximumextinction ratio over wavelengths is 40 dB and no more infinite). The values are the same for^ = 0 (Figures 36A and 36B) and ^ = 2 nm (Figures 37A and 37B).

[0260] The PDL represents the difference between the TE and TM curves. The worst-case PDL over the L2band is very low for the main output (where light is expected), at 0.008 dB, cf. Figures 37C and 37D. It can be high for the secondary output (where light is unwanted), e.g. 18 dB in L1band, cf. Figure 37B, but this PDL does not present a problem as long as the level of light stays below a specific value (to respect a target extinction ratio).

[0261] The through and cross transmission spectra (in TE and TM) of a 2x2 quad-switch (waveguide width of 2500 nm) tuned at the cross and bar states are shown in Figures 38A and 38B for a waveguide width difference ^ = 0 (no fabrication variation) and with ideal MMI 1×2 coupler ( / 0= 0.5) and ideal MMI 2×2 coupler ( / ^= 0.5). It is also shown for ^ = 2 nm (to represent fabrication variation) in Figures 39A and 39B respectively, with near realistic MMI 1×2 coupler ( / 0= 0.5) and realistic non-ideal MMI 2×2 coupler ( / ^= 0.515) in Figures40A and 40B, and both situations combined in Figures 41A and 41B. The coupling ratio value / ^ of the realistic non-ideal MMI 2×2 coupler is taken the same as the average coupling ratiovalue / \]^of the case of 2x2 switch for comparison. The coupling ratio value / 0of the near realistic MMI 1×2 coupler is kept the same as ideal MMI 1×2 coupler since this device is symmetrical. Figures 41C and 41D depict zoomed views of the main output of Figures 41A and 41B.

[0262] The worst-case extinction ratio over the L2 band is 62 dB at cross and bar states for^ = 0 and ideal couplers (Figures 38A and 38B). It remains the same when a waveguide widthdifference ^ = 2 nm is introduced (Figures 39A and 39B) thanks to the use of the large waveguide width (2500 nm). With “realistic” couplers, the extinction ratio remains almost thesame at 60 dB now (and the curves get smoother), for both ^ = 0 (Figures 40A and 40B) and^ = 2 nm (Figures 41A and 41B). The worst-case PDL over the L2 band for the main outputis 0.004 dB, cf. Figures 41C and 41D.

[0263] The inventive 2x2 quad-switch therefore demonstrates advantages over the (standard) 2x2 switch, with 2.4 times the extinction ratio (with presence of waveguide width difference and realistic couplers) and half the PDL.

[0264] Extinction Ratio

[0265] The extinction ratio of a 2x2 switch is plotted versus temperature difference U_in Figure 42, for a waveguide width of 2500 nm, a waveguide width difference ^ = 2 nm, and non-ideal MMI 2×2 couplers ( / 0= 0.51, / ^= 0.52). It is shown for TE and TM, and at two different wavelengths (1304.5 and 1317.5 nm, the limits of L2band).

[0266] The extinction ratio of a 2x2 quad-switch is plotted versus temperature difference U_in Figure 43, for a waveguide width of 2500 nm, a waveguide width difference ^ = 2 nm, and near ideal MMI 1×2 coupler ( / 0= 0.5) and non-ideal MMI 2×2 coupler ( / ^= 0.515). It is also shown for TE and TM, at the two different wavelengths. For the 2x2 switch, the extinction ratio curves are overlapping for TE and TM for the balanced cross state, but a separation is present for the bar state. For the 2x2 quad-switch a separation is present for both cross and bar states, but this separation is half the one of the 2x2 switch. These curves show the same results as in “Transmission Spectrum and PDL”, depicted in Figures 37 and 41 respectively, namely a worst-case extinction ratio over the L2 band of 25 dB for the 2x2 switch and 60 dB for the 2x2 quad-switch.

[0267] FSR

[0268] The FSR of a switch, such as 1x2 switch, 2x2 switch or 2x2 quad switch depending upon phase 1[, is displayed in Error! Reference source not found. versus waveguide width difference ^ for different target phase differences 1[(for a waveguide width of 2500 nm). For a 2x2 switch, the balanced cross state corresponds to 1[= 0 and the lowest unbalanced barstate to 1[ = ±T. For a 2x2 quad-switch the lowest unbalanced cross state corresponds to 1[ =− 3 and at the low 3^ est unbalanced bar state to 1[ = ^. The FSR of the 2x2 quad-switch is twicethe one of the 2x2 switch at ^ = 0 since it requires only half of the phase difference. It remains almost the double for moderate values of ^, even at ^ = 10 nm for example, because the stability coefficient ^^,¯is low for ^ = 2500 nm, cf. "

[0269] Expression of FSR".

[0270] Variations

[0271] The case of switches not perfectly tuned will now be considered, i.e. when applying a temperature difference U_a bit different from the temperature difference U_,YWZ[[required to reach the cross state and U_,.\Wto reach the bar state. This is to reflect situations where anelectrical micro-heater used to provide the temperature difference would have fabrication variation (e.g. in width), its properties would be ageing (e.g. electrical resistance), or theprecision of its electrical control would be limited. The temperature differences considered areU_ = U_,YWZ[[ ± 0.04U_,[ and U_ = U_,.\W ± 0.04U_,[, with U_,[ the difference of temperaturedifferences for cross and bar states as given by Equation (237). U_,[= KU_,.\W− U_,YWZ[[K. (237)

[0272] The through andnear the cross state (U_= U_,YWZ[[+ 0.04U_,[) and near the bar state (U_= U_,.\W+ 0.04U_,[) are shown in Figures 45A and 45B respectively, for a waveguide width of 2500 nm, awaveguide width difference ^ = 2 nm, and non-ideal MMI 2×2 couplers ( / 0 = 0.51, / ^ =0.52). The through and cross transmission spectra (in TE and TM) of a 2x2 quad-switch tunednear the cross state (U_= U_,YWZ[[+ 0.04U_,[) and near the bar state (U_= U_,.\W+ 0.04U_,[) are shown in Figure 46A and 46B respectively, for a waveguide width of 2500 nm, a waveguide width difference ^ = 2 nm, and near realistic MMI 1×2 coupler ( / 0= 0.5) and realistic non- ideal MMI 2×2 coupler ( / ^= 0.515).

[0273] For the 2x2 switch, at 4% difference (of U_,[) from the cross state the worst-case extinction ratio over the L2band is 23 dB, and at 4% from the bar state it is 19 dB. For the 2x2 quad-switch, at 4% from the cross and bar states the extinction ratio is 41 dB. For the other side of the 4% difference ( U_= U_,YWZ[[− 0.04U_,[and U_= U_,.\W− 0.04U_,[, not represented) the values of extinction ratios are the same.

[0274] So, the 2x2 switch extinction ratio drops from 25 dB to 19 dB when 4% untuned, and the 2x2 quad-switch extinction ratio drops from 60 dB to 41 dB when 4% untuned.

[0275] 5. OTHER SWITCH DESIGNS

[0276] Whilst the preceding sections have been focused to the design of 1x2 sub-switches and their grouping to provide a 2x2 quad-switch employing four 1x2 sub-switches the inventors have also extended the design concepts to other designs of 2x2 switch elements and other NxM switch elements.

[0277] Referring to Figure 48 there is depicted a 2x2 octo-switch employing 8 1x2 sub- switches wherein an initial pair of 1x2 sub-switches 4810A and 4810B provide 1x2 switching to first to fourth reversed 1x2 sub-switches (acting as 2x1 sub-switches) 4820A to 4820D respectively. Then these four reversed 1x2 sub-switches 4820A to 4820D respectively are coupled to a pair of reversed 1x2 sub-switches 4820E and 4820ZF respectively.

[0278] This can be further extended as depicted in Figure 49 to a 2x2 dodeca-switch employing twelve 1x2 sub-switches wherein an initial of pair 1x2 sub-switches 4910A and 4910B provide 1x2 switching to first to fourth 1x2 sub-switches 4910C to 4910F respectively. Then these are connected to four reversed 1x2 sub-switches 4820A to 4820D respectively are coupled to a pair of reversed 1x2 sub-switches (acting as 2x1 sub-switches) 4920A to 4920D respectively and therein to a final pair of reversed 1x2 sub-switches 4920E and 4920F respectively.

[0279] It is also possible to design 1xN switches such as the 1x2 tri-switch depicted in Figure 50 wherein an initial 1x2 sub-switch 5010 is coupled to a pair of reversed 1x2 sub-switches 5020A and 5020B respectively. For the cross state, the 1x2 sub-switches A and C are set on "bottom" state ("cross" state) and the 1x2 sub-switch D on "top" state (that will arbitrarily be taken as "cross" state here), thereby sending the light from port A to port D (through ports N and U). For the bar state, the 1x2 sub-switches A and C are set on "top" state ("bar" state) and the 1x2 sub-switch D on "bottom" state (that will arbitrarily be taken as "bar" state here), thereby sending the light from port A to port C (through ports M and R). The 1x2 tri-switch can be seen as a 2x2 quad-switch where the 1x2 sub-switch B is omitted. It possesses all the same advantages of performance as the 2x2 quad-switch (e.g. extinction ratio, passband), and can advantageously be used when only one input is required.

[0280] As with the 2x2 quad-switch the 1x2 tri-switch provides for configuration in an off state, where no light exits the device, or as a variable splitter and attenuator, by adjusting progressively the phase of each 1x2 sub-switches 5010A, 5020A and 5020B respectively.

[0281] It would be evident that Nx N or NxM switches can be constructed using the 2x2 quad- switch and / or 1x2 tri-switch etc. as unit cells within different architectures. Such architectures may include, but not be limited to, path-independent insertion-loss (PILOSS), crossbar, tree, Benes, Spanke-Benes, Banyan etc.

[0282] Whilst generally the description above considers a balanced MZI it would be evident that unbalanced MZIs (e.g. with one arm longer than the other) can also be used inside the 1x2 sub-switches, thereby allowing for the addition of filtering characteristics (such as elimination of other wavelengths, or inverse switching i.e. being in bar state for some wavelengths and cross state for some other wavelengths) etc.

[0283] Specific details are given in the above description to provide a thorough understanding of the embodiments. However, it is understood that the embodiments may be practiced without these specific details. For example, circuits may be shown in block diagrams in order not to obscure the embodiments in unnecessary detail. In other instances, well-known circuits,processes, algorithms, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments.

[0284] The foregoing disclosure of the exemplary embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many variations and modifications of the embodiments described herein will be apparent to one of ordinary skill in the art in light of the above disclosure. The scope of the invention is to be defined only by the claims appended hereto, and by their equivalents.

[0285] Further, in describing representative embodiments of the present invention, the specification may have presented the method and / or process of the present invention as a particular sequence of steps. However, to the extent that the method or process does not rely on the particular order of steps set forth herein, the method or process should not be limited to the particular sequence of steps described. As one of ordinary skill in the art would appreciate, other sequences of steps may be possible. Therefore, the particular order of the steps set forth in the specification should not be construed as limitations on the claims. In addition, the claims directed to the method and / or process of the present invention should not be limited to the performance of their steps in the order written, and one skilled in the art can readily appreciate that the sequences may be varied and still remain within the spirit and scope of the present invention.

Claims

CLAIMS What is claimed is:

1. A method comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range; establishing a waveguide geometry wherein a polarization phase difference within an active switching section of the interferometric optical switch comprising a difference between a phase for optical signals in a transverse magnetic (TM) mode of an optical waveguide forming the active section and another phase for optical signals in a transverse electric (TE) mode of the optical waveguide forming the active section is below a predetermined threshold over a defined range of phase differences between optical signals propagating within an arm of the interferometric optical switch wherein the defined range of phase differences is at least that to change a switch state of the interferometric optical switch to another switch state of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides.

2. The method according to claim 1, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is constant.

3. The method according to claim 1, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the opticalwaveguides forming the active switching section of the interferometric optical switch is within a defined range.

4. A method comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range; establishing a stability condition for the optical waveguides; establishing a waveguide geometry fulfilling the stability condition for an active switching section of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides. the stability condition is defined as a ratio of first rate of change to a second rate of change; the first rate of change is a differential of an effective index of a transverse magnetic (TM) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both a width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and a temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch; and the second rate of change is differential of an effective index of a transverse electric (TE) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both the width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and the temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch.

5. The method according to claim 4, wherein establishing the waveguide geometry comprises: initially establishing a set of waveguide geometries fulfilling the stability condition for the active switching section of the interferometric optical switch; establishing a value for a polarization phase difference for each waveguide geometry of the set of waveguide geometries where the polarisation phase difference ofthe interferometric optical switch comprises a difference between a phase for optical signals in a transverse magnetic (TM) mode of an optical waveguide forming the active section and another phase for optical signals in a transverse electric (TE) mode of the optical waveguide forming the active section over a defined range of phase differences between optical signals propagating within an arm of the interferometric optical switch; and selecting the waveguide geometry of the set of waveguide geometries with the minimum polarization phase difference; and the defined range of phase differences is at least that to change a switch state of the interferometric optical switch to another switch state of the interferometric optical switch.

6. The method according to claim 4, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is constant.

7. The method according to claim 4, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is within a defined range.

8. An interferometric optical switch comprising: an input optical coupler for coupling optical signals with a defined wavelength range from an input port to an end of each optical waveguide of a pair of optical waveguides; an output optical coupler coupled to a distal end of optical waveguide of the pair of optical waveguides for coupling the optical signals to an output port or another output port; and the pair of optical waveguides; whereineach optical waveguide of the pair of optical waveguides comprises a first section of a first length and a first width and a second section of a second length and a second width; the first length and the second length satisfy a condition that a first summation equals a second summation; the first summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse magnetic (TM) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TM mode of the second section of the optical waveguide; and the second summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse electric (TE) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TE mode of the second section of the optical waveguide.

9. The interferometric optical switch according to claim 8, wherein the input optical coupler is a 1x2 directional optical coupler or a 1x2 multimode interferometer; and the input optical coupler is a 2x2 directional optical coupler or a 2x2 multimode interferometer.

10. The interferometric optical switch according to claim 8, wherein the input optical coupler is a Y-junction; and the input optical coupler is a 2x2 directional optical coupler or a 2x2 multimode interferometer.

11. A method of providing an interferometric optical switch comprising providing a first 1x2 interferometric optical switch; providing a second 1x2 interferometric optical switch; providing a first 2x1 interferometric optical switch; and providing a second 2x1 interferometric optical switch; wherein a first output of the first 1x2 interferometric optical switch is coupled to an input of the first 2x1 interferometric optical switch; a second output of the first 1x2 interferometric optical switch is coupled to an input of the second 2x1 interferometric optical switch;a first output of the second 1x2 interferometric optical switch is coupled to another input of the first 2x1 interferometric optical switch; and a second output of the second 1x2 interferometric optical switch is coupled to another input of the second 2x1 interferometric optical switch.

12. The method of providing the interferometric optical switch according to claim 11, wherein each of the first 1x2 interferometric optical switch, the second 1x2 interferometric optical switch, the first 2x1 interferometric optical switch and second 2x1 interferometric optical switch comprise a pair of optical waveguides coupled at one end to an input coupler and at another end to an output coupler; each optical waveguide of the pair of optical waveguides comprises a first section of a first length and a first width and a second section of a second length and a second width; the first length and the second length satisfy a condition that a first summation equals a second summation; the first summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse magnetic (TM) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TM mode of the second section of the optical waveguide; and the second summation being the sum of the length of the first section multiplied by a rate of change of the effective index of a transverse electric (TE) mode of the first section of the optical waveguide and the length of the second section multiplied by a rate of change of the effective index of a TE mode of the second section of the optical waveguide.

13. The method of providing the interferometric optical switch according to claim 11, wherein each of the first 1x2 interferometric optical switch, the second 1x2 interferometric optical switch, the first 2x1 interferometric optical switch and second 2x1 interferometric optical switch comprise a pair of optical waveguides coupled at one end to an input coupler and at another end to an output coupler; and the pair of optical waveguides were designed using a process comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range;establishing a waveguide geometry wherein a polarization phase difference within an active switching section of the interferometric optical switch comprising a difference between a phase for optical signals in a transverse magnetic (TM) mode of an optical waveguide forming the active section and another phase for optical signals in a transverse electric (TE) mode of the optical waveguide forming the active section is below a predetermined threshold over a defined range of phase differences between optical signals propagating within an arm of the interferometric optical switch wherein the defined range of phase differences is at least that to change a switch state of the interferometric optical switch to another switch state of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides.

14. The method according to claim 13, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is constant.

15. The method according to claim 13, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is within a defined range.

16. The method of providing the interferometric optical switch according to claim 11, wherein each of the first 1x2 interferometric optical switch, the second 1x2 interferometric optical switch, the first 2x1 interferometric optical switch and second 2x1 interferometricoptical switch comprise a pair of optical waveguides coupled at one end to an input coupler and at another end to an output coupler; and the pair of optical waveguides were designed using a process comprising: establishing a material system for forming an interferometric optical switch employing optical waveguides operating within a defined wavelength range; establishing a stability condition for the optical waveguides; establishing a waveguide geometry fulfilling the stability condition for an active switching section of the interferometric optical switch; and at least one of designing and fabricating the interferometric optical switch in dependence upon the established waveguide geometry; wherein a width of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is either larger than or smaller than a thickness of that optical waveguide of the optical waveguides. the stability condition is defined as a ratio of first rate of change to a second rate of change; the first rate of change is a differential of an effective index of a transverse magnetic (TM) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both a width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and a temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch; and the second rate of change is differential of an effective index of a transverse electric (TE) mode of the subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to both the width of the subset of the optical waveguides forming the active switching section of the interferometric optical switch and the temperature of the subset of the optical waveguides forming the active switching section of the interferometric optical switch.

17. The method of providing the interferometric optical switch according to claim 16, wherein establishing the waveguide geometry comprises: initially establishing a set of waveguide geometries fulfilling the stability condition for the active switching section of the interferometric optical switch;establishing a value for a polarization phase difference for each waveguide geometry of the set of waveguide geometries where the polarisation phase difference of the interferometric optical switch comprises a difference between a phase for optical signals in a transverse magnetic (TM) mode of an optical waveguide forming the active section and another phase for optical signals in a transverse electric (TE) mode of the optical waveguide forming the active section over a defined range of phase differences between optical signals propagating within an arm of the interferometric optical switch; and selecting the waveguide geometry of the set of waveguide geometries with the minimum polarization phase difference; and the defined range of phase differences is at least that to change a switch state of the interferometric optical switch to another switch state of the interferometric optical switch.

18. The method according to claim 16, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is constant.

19. The method according to claim 16, wherein a derivative of the effective refractive index of a waveguide mode of each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch with respect to the width of the each optical waveguide of a subset of the optical waveguides forming the active switching section of the interferometric optical switch is within a defined range.

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