Photonic integrated circuit and operation method for gradient-based tunable echelle grating mux / demuxes

Abstract

The embodiment relates to a photonic integrated circuit including an tunable echelle grating multiplexer / demultiplexer. This tunable echelle grating multiplexer / demultiplexer is configured to selectively generate a monotonic refractive index gradient across a free propagation region during operation to adjust the peak transmission wavelength(s) of one or more channels of the echelle grating multiplexer / demultiplexer. Compared to conventional tunable echelle grating multiplexers / demultiplexers, the tunable echelle grating multiplexer / demultiplexer described herein can adjust these peak transmission wavelengths in a power-efficient manner.

Description

Gradient-based tunable echelle grating multiplexer / demultiplexer The present disclosure generally relates to echelle grating multiplexers / demultiplexers and photonic integrated circuits incorporating same that can be tuned using controllable refractive index gradients. Echelle grating multiplexers / demultiplexers play an important role in the operation of photonic integrated circuits because they allow the efficient combination or separation of multiple wavelengths of light carried by the waveguides of the photonic integrated circuit. The operation of an echelle grating multiplexer / demultiplexer is highly sensitive to changes in the effective refractive index of the medium that propagates light between its input waveguide(s) and its output waveguide(s). Unintended shifts in this effective refractive index can result in power loss or backreflections into the input waveguides, each of which can adversely affect the operation of the photonic integrated circuit. Accordingly, it may be desirable to provide an echelle grating multiplexer / demultiplexer that is adjustable to account for unintended changes in the effective refractive index. Embodiments described herein relate to a photonic integrated circuit including a tunable echelle grating multiplexer / demultiplexer. In some embodiments, the echelle grating multiplexer / demultiplexer comprises: a plurality of waveguides; an echelle grating; a free propagation region positioned between the echelle grating and the plurality of waveguides; and a first heater positioned to generate a first monotonic temperature gradient across the free propagation region. The plurality of waveguides may include a set of input waveguides and a set of output waveguides. In some variations, the first heater does not overlap the free propagation region. In some variations, the photonic integrated circuit includes a slab waveguide forming the free propagation region, and the first heater comprises a first doped region of the slab waveguide. Additionally or alternatively, the echelle multiplexer / demultiplexer includes a second heater positioned to generate a second monotonic temperature gradient across the free propagation region. In some of these variations, the first heater and the second heater are positioned on opposite sides of the free propagation region. In some of these variations, the first heater and the second heater are positioned on a common side of the free propagation region. The second heater may be positioned so that it does not overlap the free propagation region. In other embodiments, an echelle grating multiplexer / demultiplexer includes: a plurality of waveguides having a set of input waveguides and a set of output waveguides; an echelle grating; and a free propagation region positioned between the echelle grating and the plurality of waveguides. In these embodiments, the photonic integrated circuit includes: a light source unit optically connected to the set of input waveguides; and a controller configured to measure a light intensity using a power monitor optically connected to at least one of the set of output waveguides. The controller is configured to control the echelle grating multiplexer / demultiplexer based on the measured light intensity to generate a monotonic refractive index gradient across the free propagation region. In some variations, the echelle multiplexer / demultiplexer includes a first heater positioned on a first side of the free propagation region, and the controller is operably connected to the first heater. The first heater can be controlled to produce the monotonic refractive index gradient. The photonic integrated circuit can include a slab waveguide forming the free propagation region, and the first heater can include a first doped region of the slab waveguide. In some variations, the echelle multiplexer / demultiplexer further includes a second heater positioned on the first side of the free propagation region, and the controller is operably connected to the second heater. In other variations, the echelle multiplexer / demultiplexer further includes a second heater positioned on a second side of the free propagation region opposite the first side. The controller can control the operation of the second heater to produce the monotonic refractive index gradient. The properties of the monotonic refractive index gradient can vary depending on the heater currently operating and the operating parameters of the operating heaters (e.g., how much heat each heater is generating). Additionally or alternatively, the echelle grating multiplexer / demultiplexer includes a first force applicator positioned to selectively apply a first force on a first side of the free propagation region, and the controller is operably connected to the first force applicator. The controller can control the first force applicator to apply the first force at the first side of the free propagation region and thereby generate the monotonic refractive index gradient. In some variations, the echelle grating multiplexer / demultiplexer includes a second force applicator, such that the controller is operably connected to the second force applicator. The second force applicator can be positioned to selectively apply a second force on a second side of the free propagation region, and the. In other variations, the second force applicator can be positioned to selectively apply a second force on the first side of the free propagation region. The controller can control the second force applicator to apply the second force and thereby generate the monotonic refractive index gradient. The properties of the monotonic refractive index gradient may vary depending on the force applicator currently operating and the magnitude of the force(s) being applied by the force applicator(s) currently operating. Other embodiments relate to a method of operating an echelle grating multiplexer / demultiplexer, wherein the echelle grating multiplexer / demultiplexer comprises a set of input waveguides, a set of output waveguides, an echelle grating, and a free propagation region. The method may include generating a first monotonic refractive index gradient across the free propagation region during a first time period, the first monotonic refractive index gradient decreasing along a direction. The method further includes generating a second monotonic refractive index gradient across the free propagation region during a second time period, the second monotonic refractive index gradient increasing along the direction. Accordingly, the method can correct peak wavelength shifts in different directions relative to a target peak wavelength shift of a channel of the echelle grating multiplexer / demultiplexer at different times. In some variations, the method includes, prior to the first time period, detecting a first peak transmission wavelength shift in a channel of the echelle grating multiplexer / demultiplexer relative to a target peak transmission wavelength, wherein the first monotonic refractive index gradient is generated in response to detecting the first peak transmission wavelength shift. In some of these variations, prior to the second time period, detecting a second peak transmission wavelength shift in the channel of the echelle grating multiplexer / demultiplexer relative to the target peak transmission wavelength, wherein the second monotonic refractive index gradient is generated in response to detecting the second peak transmission wavelength shift. In some cases, generating the first monotonic refractive index gradient during the first time period includes generating a first monotonic temperature gradient across the free propagation region using a first heater positioned on a first side of the free propagation region, the first monotonic temperature gradient decreasing in the direction. In some of these variations, a slab waveguide forms the free propagation region, and the first heater comprises a first doped region of the slab waveguide. Additionally or alternatively, generating the second monotonic refractive index gradient during the second time period may include generating a second monotonic temperature gradient across the free propagation region using a second heater positioned on a second side of the free propagation region, the second monotonic temperature gradient increasing in the direction. In some of these variations, the second heater comprises a second doped region of the slab waveguide. In other variations, generating the first monotonic refractive index gradient during the first time period includes applying a first force on a first side of the free propagation region to generate a first monotonic stress gradient across the free propagation region, the first monotonic stress gradient increasing along the direction. In some of these variations, generating the second monotonic refractive index gradient during the second time period includes applying a second force on a second side of the free propagation region to generate a second monotonic stress gradient across the free propagation region, the second monotonic stress gradient decreasing along the direction. In addition to the example aspects and embodiments described herein, further aspects and embodiments will become apparent by reference to the drawings and by study of the following descriptions. [Cross-reference to related applications] This application is a non-provisional application and claims priority under 35 USC § 119(e) to U.S. Provisional Patent Application No. 63 / 615,107, filed on December 27, 2023, the contents of which are incorporated herein by reference as if fully set forth herein. Reference will now be made in detail to representative embodiments illustrated in the accompanying drawings. It should be understood that the following description is not intended to limit the embodiments to a single preferred embodiment. On the contrary, the intention is to cover various alternatives, modifications, and equivalents that may be included within the spirit and scope of the described embodiments as defined by the appended claims. Disclosed below is a photonic integrated circuit including a tunable échelle grating multiplexer / demultiplexer. The tunable échelle grating multiplexer / demultiplexer described herein is configured to selectively generate a monotonic refractive index gradient across a free propagation region during operation to adjust the peak transmission wavelength(s) of one or more channels of the échelle grating multiplexer / demultiplexer. When the monotonic refractive index gradient is generated across the free propagation region, the effective refractive index of the free propagation region will vary across the free propagation region (e.g., different portions of the free propagation region will have different corresponding effective refractive indices). Compared to conventional tunable échelle grating multiplexers / demultiplexers, the tunable échelle grating multiplexer / demultiplexer described herein can adjust these peak transmission wavelengths in a power-efficient manner. The tunable echelle grating multiplexer / demultiplexer can be operably connected to a controller that can control the tunable echelle grating multiplexer / demultiplexer to generate a monotonic refractive index gradient across the free propagation region. For example, the controller can measure light intensity using a power monitor optically connected to at least one of the output waveguides of the tunable echelle grating multiplexer / demultiplexer (e.g., by dropping a portion of the light carried by one or more output waveguides) and use the measured intensity as feedback to control the generation of the monotonic refractive index gradient across the free propagation region. Depending on the photonic integrated circuit into which the echelle grating multiplexer / demultiplexer is incorporated, the tunable echelle grating multiplexer / demultiplexer described herein can be operated as a multiplexer (e.g., to simultaneously or sequentially route light of different wavelengths from multiple input waveguides to a common output waveguide) or a demultiplexer (e.g., to simultaneously or sequentially route light of multiple different wavelengths from a common input waveguide to multiple output waveguides). These and other embodiments are discussed below with reference to Figures 1A through 3. However, those skilled in the art will appreciate that the detailed description given herein with respect to these figures is for illustrative purposes only and should not be construed as limiting. FIG1A shows a top view of a photonic integrated circuit 100 including an echelle grating multiplexer / demultiplexer 102, and FIG1B shows a cross-sectional side view thereof (taken along line 1B-1B). The echelle grating multiplexer / demultiplexer 102 has a plurality of waveguides, including a set (e.g., one or more) of input waveguides (shown as a single input waveguide 104 in FIG1A ) and a set (e.g., one or more) of output waveguides. For example, in the variation shown in FIG1A , the set of input waveguides includes a single input waveguide 104, and the set of output waveguides includes a plurality of output waveguides 106 a through 106 c. The echelle grating multiplexer / demultiplexer 102 further includes an echelle grating 108 positioned to redirect light received from the one or more input waveguides to the one or more output waveguides. A free propagation region 110 is positioned between the echelle grating 108 and the plurality of waveguides. Specifically, the free propagation region 110 is positioned between the echelle grating 108 and one or more input waveguides (e.g., input waveguide 104), and also between the echelle grating 108 and one or more output waveguides (e.g., the plurality of output waveguides 106a-106c). Light can be introduced into the free propagation region 110 from an input waveguide (e.g., input waveguide 104) and can traverse the free propagation region 110 until it reaches the echelle grating 108. The echelle grating 108 is made of a reflective material (e.g., reflective at the wavelength of the light carried by the echelle grating multiplexer / demultiplexer 102) and can include a groove pattern that diffracts incident light from the input waveguide as it is redirected toward the set of output waveguides. The echelle grating 108 disperses light based on its wavelength, and thus the light coupled between a particular input waveguide and a particular output waveguide depends on the wavelength of the light launched from that input waveguide into the free propagation region 110. While the echelle grating multiplexer / demultiplexer 102 of FIG. 1A is shown as being configured as a demultiplexer having a single input waveguide 104 and a plurality of output waveguides 106 a to 106 c, it should be understood that in other cases, the echelle grating multiplexer / demultiplexer 102 can be configured as a multiplexer having a plurality of input waveguides and a single output waveguide. In still other variations, the echelle grating multiplexer / demultiplexer 102 can be configured as a multiplexer or demultiplexer having a plurality of input waveguides and a plurality of output waveguides. The principles described herein can be applied to any echelle grating multiplexer / demultiplexer 102 capable of routing light between at least one input waveguide and at least one output waveguide. The set of input waveguides, the set of output waveguides, and the free propagation region 110 can be formed by corresponding portions of a waveguide layer of the photonic integrated circuit 100. Specifically, the photonic integrated circuit 100 can include a substrate 112 and a set of layers supported by the substrate 112. The set of layers includes a first cladding layer 114 and a waveguide layer 116 positioned on the first cladding layer 114 such that the first cladding layer 114 is positioned between the waveguide layer 116 and the substrate 112. The waveguide layer 116 can be patterned or otherwise formed to define any input waveguides, output waveguides, and the free propagation region 110 of the echelle grating multiplexer / demultiplexer 102. For example, the waveguide layer 116 can form a slab waveguide between the echelle grating 108 and the input waveguides 104 and the output waveguides 106a-106c, such that a portion of the slab waveguide serves as the free propagation region 110. For the purposes of this discussion, the free propagation region 110 is considered to include the portion of the waveguide layer 116 (e.g., the portion of a slab waveguide) through which light can travel to pass through any pair of input and output waveguides. In other words, during operation of the echelle multiplexer / demultiplexer 102, the portion of the waveguide layer outside the free propagation region 110 will not receive light passing between any input waveguide and any output waveguide of the echelle multiplexer / demultiplexer 102. The first cladding layer 114 can provide vertical optical confinement for light traveling through the waveguide layer 116 (e.g., for light carried by the echelle multiplexer / demultiplexer 102). In some cases, one or more additional surfaces of the waveguide layer 116 can be covered by a second cladding layer 118, which can also provide vertical optical confinement for light traveling through the waveguide layer 116. In some cases, the photonic integrated circuit 100 can include a third cladding layer 120 coplanar with the waveguide layer 116, which can be used to define the boundaries of the input and output waveguides of the echelle multiplexer / demultiplexer 102 (and thereby provide horizontal optical confinement for light traveling through the waveguide layer 116). For example, a portion of the third cladding layer 120 shown in FIG. 1B can partially define the boundaries of the second output waveguide 106b of FIG. 1A. Although the second and third cladding layers 118, 120 are shown in FIG. 1B as separate layers, it should be understood that the second cladding layer 118 and the third cladding layer 120 may be formed as different portions of the same layer (eg, as part of a common processing step). The various layers of the photonic integrated circuit 100 can be formed from any suitable material, depending on the wavelength or wavelengths of light to be carried by the waveguide layer 116. For example, in some variations, the photonic integrated circuit 100 is configured to carry infrared light of one or more wavelengths. In some of these variations, the waveguide layer 116 is formed from silicon, silicon nitride, silicon dioxide, or the like, the first cladding layer 114, the second cladding layer 118, and the third cladding layer 120 are formed from one or more dielectric materials such as silicon dioxide, and the substrate 112 is formed from silicon. In some of these cases, the photonic integrated circuit 100 can be fabricated using silicon-on-insulator technology. An echelle grating multiplexer / demultiplexer is a wavelength-specific structure such that the amount of light coupled between a given input waveguide and a given output waveguide depends on the wavelength of the light introduced into the input waveguide. Accordingly, each pair of input and output waveguides of the echelle grating multiplexer / demultiplexer (also referred to herein as a "channel" of the echelle grating multiplexer / demultiplexer) has a peak transmission wavelength, which represents the wavelength of light with maximum transmission between the input and output waveguides. Accordingly, depending on the specific wavelength being multiplexed or demultiplexed, the echelle grating multiplexer / demultiplexer 102 can be designed to have a specific corresponding peak transmission wavelength for each channel of the echelle grating multiplexer / demultiplexer 102. For example, the echelle grating multiplexer / demultiplexer 102 can be designed to have a target peak transmission wavelength for a first channel. The photonic integrated circuit 100 can be configured to generate light at a target peak transmission wavelength and transmit the light to the echelle multiplexer / demultiplexer 102 to route the light between the input waveguide and the output waveguide of the first channel. In practice, however, the actual peak transmission wavelength of a given channel is highly dependent on the refractive index of the free propagation region 110. For example, manufacturing variations (e.g., variations in the thickness of the waveguide layer 116, stresses applied to the waveguide layer 116 during fabrication or packaging of the photonic integrated circuit 100) may cause these peak transmission wavelengths to deviate from the expected target peak transmission wavelengths for the channels of the echelle multiplexer / demultiplexer 102. This can lead to performance differences across different devices. Similarly, within a given device, the performance of the echelle multiplexer / demultiplexer may vary over time due to variations in operating temperature and / or local stresses that may be applied to the echelle multiplexer / demultiplexer. To correct for these performance variations, an echelle grating multiplexer / demultiplexer can be configured to have a free propagation region with a tunable refractive index. Generally, such tunable echelle grating multiplexers / demultiplexers are configured with a heater positioned above the free propagation region. The heater is operated to uniformly heat the free propagation region (and thereby uniformly change the effective refractive index of the free propagation region), which can thereby change the peak transmission wavelength of the various channels of the echelle grating multiplexer / demultiplexer. One such example is disclosed in U.S. Patent No. 11,561,346 B2, entitled "Tunable Echelle Grating," the entire contents of which are hereby incorporated by reference. Such heaters have relatively large power consumption requirements compared to the tunable echelle grating multiplexers / demultiplexers described herein. Additionally, because the heater can only change the refractive index in a single direction, it may be necessary to design such echelle multiplexers / demultiplexers for worst-case operating conditions, so that the heater is always operational to achieve the desired peak transmission wavelength for a given channel. This can further increase the average power consumption of the echelle multiplexer / demultiplexer during typical operating conditions. In contrast, the tunable échelle grating multiplexer / demultiplexer described herein is configured to produce a monotonic refractive index gradient across its free propagation region. As used herein, a "monotonic" gradient (e.g., monotonic refractive index gradient, monotonic temperature gradient) refers to a gradient whose value is monotonically increasing (e.g., not decreasing) or monotonically decreasing (e.g., not increasing) along a particular direction. A non-uniform variation in the effective refractive index across the free propagation region of an échelle grating multiplexer / demultiplexer may produce a shift in the peak transmission wavelength of a channel. This shift may require a smaller maximum effective refractive index change in the free propagation region than would be required for a tunable échelle grating multiplexer / demultiplexer that produces a uniform variation in the effective refractive index across the free propagation region. This can reduce the power required to produce a given shift in the peak transmission wavelength of a channel. Additionally, some variations of the tunable échelle grating multiplexer / demultiplexer described herein can be configured to selectively produce a monotonic refractive index gradient increase in different directions across their free propagation region. This can allow the tunable echelle grating multiplexer / demultiplexer to selectively shift the peak transmission wavelength in a first direction during a first time period (e.g., to reduce the peak transmission wavelength of a given channel), or shift the peak transmission wavelength in a second direction during a second time period (e.g., to increase the peak transmission wavelength of a given channel). This can allow the design of the tunable echelle grating multiplexer / demultiplexer to be designed for typical operating conditions, as the tunable echelle grating multiplexer / demultiplexer can be capable of correcting for peak transmission wavelength shifts in either direction. In some variations, a tunable echelle multiplexer / demultiplexer as described herein may utilize one or more heaters, each of which can be used to generate a corresponding monotonic temperature gradient across the free propagation region. The monotonic temperature gradient generated in this manner can produce a monotonic refractive index gradient across the free propagation region. For example, FIG2A shows a top view of a variation of a photonic integrated circuit 200 including a tunable echelle multiplexer / demultiplexer 201. The tunable echelle multiplexer / demultiplexer 201 can be configured as described above with respect to the echelle multiplexer / demultiplexer 102 of FIG1A and FIG1B , except that the tunable echelle multiplexer / demultiplexer 201 includes a heater 202. In these variations, the heater 202 is controllable to selectively generate heat. The heater 202 is positioned relative to the free propagation region 110 such that the heat generated by the heater 202 generates a monotonic temperature gradient across the free propagation region 110. For example, as shown in FIG2A , heater 202 is positioned on a first side of free propagation region 110. When heater 202 is operated to generate heat, the heat generated by heater 202 will be distributed across free propagation region 110 (e.g., through the portion of waveguide layer 116 that forms free propagation region 110). Free propagation region 110 is not uniformly heated by heater 202, and therefore the temperature of free propagation region 110 will decrease along first gradient direction 208, as shown in FIG2A . In this manner, heater 202 can generate a monotonic temperature gradient across free propagation region 110 that decreases along first gradient direction 208 (e.g., the temperature on a first side of free propagation region 110 is higher than the temperature on a second side of free propagation region 110 relative to the first side). Because the effective refractive index of the free propagation region 110 depends on the temperature of the free propagation region 110, the monotonic temperature gradient generated by the heater 202 will also generate a monotonic refractive index gradient across the free propagation region 110. In this case, the monotonic refractive index gradient decreases along the first gradient direction 208. In other words, the effective refractive index of the free propagation region 110 decreases from a first side of the free propagation region 110 to a second side of the free propagation region 110. This non-uniform refractive index variation can be used to adjust the peak transmission wavelength of one or more channels of a tunable echelle grating multiplexer / demultiplexer. For example, in the variation shown in FIG2A , the input waveguide 104 can be optically connected to a light source unit 204 configured to generate light of a specific wavelength or range of wavelengths, such that the input waveguide 104 receives light from the light source unit 204. The light source unit 204 includes a set of light sources (not shown), each of which is selectively operable to emit light at a corresponding set of wavelengths. Each light source can be any component capable of generating light at one or more specific wavelengths, such as a light-emitting diode or a laser. A laser can include a semiconductor laser, such as a laser diode (e.g., a distributed Bragg reflector laser, a distributed feedback laser, an external cavity laser), a quantum cascade laser, or the like. A given light source can be single-frequency (fixed wavelength) or tunable to selectively generate one of multiple wavelengths (i.e., the light source can be controlled to selectively output different wavelengths at different times). The set of light sources can include any suitable combination of light sources and can collectively operate to generate light at any of a plurality of different wavelengths. If the light source unit 204 is capable of generating multiple different wavelengths, the light source unit 204 can be configured to generate light of different wavelengths simultaneously and / or sequentially. The light source unit 204 can be integrated into the photonic integrated circuit 200, or can be separate from the photonic integrated circuit 200 and couple light into the photonic integrated circuit 200. In variations where the tunable echelle grating multiplexer / demultiplexer 201 includes multiple input waveguides 104, different input waveguides can receive light (e.g., from the light source unit 204) simultaneously or at different times, depending on the operation of the photonic integrated circuit 200. Light source unit 204 generates light that is introduced into free propagation region 110 through input waveguide 104, and at least a portion of this light is received by a corresponding output waveguide of tunable echelle grating multiplexer / demultiplexer 201. The amount of light transferred between the input waveguide and the output waveguide depends on the wavelength of the light and the peak transmission wavelength of the channel. Accordingly, it may be desirable to adjust the peak transmission wavelength of a particular channel to a target peak transmission wavelength during operation of tunable echelle grating multiplexer / demultiplexer 201. For example, the tunable echelle multiplexer / demultiplexer 201 shown in FIG2A includes a controller 206 operably connected to a heater 202. Controller 206 can control heater 202 to selectively generate heat and thereby adjust the peak transmission wavelength of a channel of echelle multiplexer / demultiplexer 201. In some of these variations, controller 206 can operate as a closed-loop system, wherein controller 206 measures the amount of light transferred for a given channel. Specifically, a portion of the light received by a channel's output waveguide (e.g., output waveguide 106c in FIG2A ) can be tapped from the output waveguide and measured by controller 206 (e.g., using an optical power monitor optically connected to at least one of the set of output waveguides). The measured portion of light can be used as a feedback signal to control heater 202 to selectively generate a monotonic temperature gradient across free propagation region 110. The controller 206 may include any combination of software, hardware, and firmware as needed to perform these functions (including any method steps described below), including, for example, one or more processors and / or application-specific integrated circuits (ASICs). Although a single controller 206 is shown in FIG2A , it should be understood that the functions of the controller described herein may be distributed across different components (e.g., different processors) that may work together to perform the functions of the controller. FIG2B through FIG2D illustrate variations of a heater 202 that can be used with the tunable echelle grating multiplexer / demultiplexer described herein. Specifically, FIG2B shows a cross-sectional view of a first variation of a photonic integrated circuit 220 (taken along line 2B-2B of the photonic integrated circuit 200 in FIG2A ), in which the heater 202 is coplanar with the waveguide layer 116 that forms the free propagation region 110. In some of these variations, the heater 202 can be formed from a doped region 222 of the waveguide layer 116, such that the doped region 222 has a higher conductivity than surrounding portions of the waveguide layer 116. In other words, a slab waveguide can form the free propagation region, and the heater 202 can be formed from the doped region 222 of the slab waveguide. For example, if the waveguide layer 116 is formed of silicon, the waveguide layer 116 can be doped with a Group III or Group V material to form the doped region 222. Although shown in FIG. 2B as extending completely through the waveguide layer 116 (e.g., from the top surface of the waveguide layer to the bottom surface of the waveguide layer), the doped region 222 may alternatively extend only partially through the waveguide layer 116 (e.g., only a portion of the thickness of the waveguide layer 116 is doped to form the doped region 222). In variations where the photonic integrated circuit 220 includes the second cladding layer 118 positioned on the waveguide layer, the photonic integrated circuit 220 may include a set of vias 224 extending through the second cladding layer 118. The set of vias 224 may provide electrical connections between the doped regions 222 and the controller 206. To generate heat using the heater 202, the controller 206 may drive current through the doped regions 222 (e.g., via the set of vias 224). In the case where the heater 202 is coplanar with the waveguide layer 116, the heater 202 can directly heat the waveguide layer 116 (e.g., as opposed to indirectly heating the waveguide layer 116 through another layer, such as the second cladding layer 118). This can reduce the power required to generate a specific monotonic temperature gradient across the free propagation region 110. Additionally, in variations where the heater 202 is coplanar with the waveguide layer 116, it may be desirable to position the heater 202 so that it does not overlap with the free propagation region 110. In other words, the doped region 222 is positioned outside the free propagation region 110. In such cases, light passing through the free propagation region 110 during operation of the echelle multiplexer / demultiplexer 201 will not interact with the heater 202 (e.g., with the doped region 222 of the slab waveguide), which could otherwise cause optical losses. In other variations, the heater may be positioned above the waveguide layer 116. For example, FIG2C shows a cross-sectional view of another variation of a photonic integrated circuit 230 (taken along line 2B-2B of the photonic integrated circuit 200 in FIG2A ) in which the heater 202 is positioned on the top surface of the waveguide layer 116. In such variations, a conductive material 232 (e.g., gold, nickel, titanium, chromium, etc.) forming the heater 202 may be deposited or otherwise positioned on the top surface of the waveguide layer 116 such that the conductive material 232 is in direct contact with the waveguide layer 116. If the photonic integrated circuit 230 includes the second cladding layer 118, the conductive material 232 may be coplanar with the second cladding layer 118. To operate the heater 202, the controller 206 may drive a current through the conductive material 232, thereby generating heat. In other variations, the heater may be positioned atop the second cladding layer 118. For example, FIG2D shows a cross-sectional view of another variation of a photonic integrated circuit 240 (taken along line 2B-2B of the photonic integrated circuit 200 in FIG2A ) in which the heater 202 is positioned on the top surface of the second cladding layer 118. In such variations, a conductive material 242 (e.g., gold, nickel, titanium, chromium, etc.) may be deposited or otherwise positioned on the top surface of the second cladding layer 118. In some variations, a portion 244 of the second cladding layer 118 may be thinned relative to surrounding portions of the second cladding layer 118, such that the conductive material 242 (and, therefore, the heater 202) is positioned closer to the waveguide layer 116. To operate the heater 202, the controller 206 may drive current through the conductive material 242, thereby generating heat. In some cases where the heater 202 is not coplanar with the waveguide layer 116, the heater may be positioned to slightly overlap the free propagation region 110, as long as the overlap is small enough to still allow the heater 202 to produce a monotonic temperature gradient across the free propagation region 110. In such cases, a portion of the heater 202 may be positioned above the light traversing the free propagation region 110. In other cases, the heaters 202 shown in Figures 2C and 2D may also be positioned so that they do not overlap the free propagation region 110 (e.g., the entire heater 202 is positioned to one side of the boundary of the free propagation region). While the tunable echelle multiplexer / demultiplexer 201 of FIG. 2A includes a single heater, it should be understood that in other cases, the tunable echelle multiplexer / demultiplexer 201 may include multiple heaters. For example, FIG. 2E shows a variation of the photonic integrated circuit 250 that includes an echelle multiplexer / demultiplexer 251 with multiple heaters. The photonic integrated circuit 250 and the echelle multiplexer / demultiplexer 251 can be configured identically to the photonic integrated circuit 200 and the echelle multiplexer / demultiplexer 201 of FIG. 2A , except that heater 202 has been replaced with a first heater 252 a and a second heater 252 b. In this variation, the first heater 252 a and the second heater 252 b are positioned on a common side of the free propagation region 110 and can be independently controlled by the controller 206. One or both heaters 252a, 252b can be operated to produce a monotonic temperature gradient that decreases in one direction. For example, during a first time period, a single heater (e.g., first heater 252a) can be operated to generate heat and thereby produce a first monotonic temperature gradient, wherein the temperature decreases in that direction. During a second time period, both heaters 252a, 252b can be operated simultaneously to generate heat and thereby produce a second monotonic temperature gradient, also decreasing in that direction. A single heater can be operated to produce a smaller peak transmission wavelength shift in a given direction, and both heaters can be operated simultaneously to produce a larger peak transmission wavelength shift in the same direction. In other cases, the first and second heaters 252a, 252b can be positioned on the same side of the free propagation region and controlled together, thereby treating heaters 252a, 252b as a single heater having two spaced-apart segments. Although shown as having the same size in FIG2E , it should be understood that the first heater 252 a and the second heater 252 b may have different sizes (e.g., the first heater 252 a may be larger than the second heater 252 b). Additionally, the first heater 252 a and the second heater 252 b may be configured in any suitable manner, such as described with respect to FIG2B through FIG2D . For example, in some variations, the first heater 252 a may be formed from a first doped region of the slab waveguide (e.g., on a first side of the free propagation region 110), and the second heater 252 b may be formed from a second doped region of the slab waveguide (e.g., also on a first side of the free propagation region 110). Additionally or alternatively, the tunable echelle multiplexer / demultiplexer described herein may include multiple heaters positioned on opposite sides of the free propagation region. For example, FIG2F shows a variation of a photonic integrated circuit 250 including an echelle multiplexer / demultiplexer 251, configured identically to the photonic integrated circuit 200 and echelle multiplexer / demultiplexer 201 of FIG2A, except that heater 202 has been replaced with a first heater 262a and a second heater 262b. In this variation, the first heater 262a and the second heater 262b are positioned on opposite sides of the free propagation region 110 and are independently controlled by a controller 206. The first and second heaters 262a, 262b are operable to generate monotonic temperature gradients that decrease in opposite directions across the free propagation region 110. Specifically, first heater 262a is positioned on a first side of free propagation region 110 and is operable to increase the temperature on the first side of free propagation region 110 relative to the temperature on an opposite, second side of free propagation region 110. This can produce a first monotonic temperature gradient that decreases along a first gradient direction 208 across free propagation region 110 during certain time periods. Second heater 262b is positioned on a second side of free propagation region 110, opposite the first side, and is operable to increase the temperature on the second side of free propagation region 110 relative to the first side of free propagation region 110. This can produce a second monotonic temperature gradient that decreases along a second gradient direction 268 across the free propagation region during other time periods. Accordingly, depending on which heater is generating heat, echelle multiplexer / demultiplexer 261 can be operated to selectively produce a monotonic temperature gradient that increases or decreases along a given direction. The first heater 262a and the second heater 262b can be configured in any suitable manner, such as described with respect to FIG. 2B through FIG. 2D . For example, in some variations, the first heater 262a can be formed from a first doped region of the slab waveguide (e.g., on a first side of the free propagation region 110), and the second heater 262b can be formed from a second doped region of the slab waveguide (e.g., positioned on a second side of the free propagation region 110). The first heater 262a and the second heater 262b can be positioned so that each does not overlap the free propagation region 110. The temperature gradients generated by the first and second heaters 262a, 262b can be used to adjust the peak transmission wavelength of a given channel in multiple directions at different times, thereby allowing the peak transmission wavelength to be selectively increased or decreased. For example, during a first time period, the echelle multiplexer / demultiplexer 261 can be operated to generate a first monotonic refractive index gradient across the free propagation region that decreases along a direction (e.g., along the first gradient direction 208 in FIG. 2F ). Specifically, the first heater 262a can be operated during the first time period to generate a first monotonic temperature gradient that decreases along the direction. Prior to the first time period, the controller 206 can detect (e.g., using measurement light from an optical power monitor as feedback) a first peak transmission wavelength shift relative to a target peak transmission wavelength in the channel of the echelle multiplexer / demultiplexer 261. In response to detecting the first peak transmission wavelength shift, the controller 206 can control the echelle multiplexer / demultiplexer 261 to generate the first monotonic refractive index gradient, which can shift the peak transmission wavelength back to the target peak transmission wavelength. In this manner, the first heater 262a may be used to correct a peak transmission wavelength shift occurring in a first direction (eg, red-shift the peak transmission wavelength toward a target peak transmission wavelength). Conversely, during a second time period, the echelle multiplexer / demultiplexer 261 may be operated to generate a second monotonic refractive index gradient across the free propagation region, the second monotonic refractive index gradient increasing along the direction (e.g., along the first gradient direction 208 in FIG. 2F ). Specifically, the second heater 262b may be operated during the second time period to generate a second monotonic temperature gradient that increases along the direction and decreases along the opposite direction (e.g., along the second gradient direction 268 in FIG. 2F ). Prior to the second time period, the controller 206 may detect (e.g., using measurement light from an optical power monitor as feedback) a second peak transmission wavelength shift relative to a target peak transmission wavelength in the channels of the echelle multiplexer / demultiplexer 261. The second peak transmission wavelength shift may be in a second direction opposite to the first direction of the first peak transmission wavelength shift. In response to detecting the second peak transmission wavelength shift, the controller 206 may control the echelle multiplexer / demultiplexer 261 to generate a second monotonic refractive index gradient, which may shift the peak transmission wavelength from the second direction back to the target peak transmission wavelength. Accordingly, the first heater 262a can be used to correct the peak transmission wavelength shift in the first direction (e.g., redshifting the peak transmission wavelength toward the target peak transmission wavelength), and the second heater 262b can be used to correct the peak transmission wavelength shift in the second direction (e.g., blueshifting the peak transmission wavelength toward the target peak transmission wavelength). In addition to or in lieu of using a heater to generate a monotonic refractive index gradient, variations of the echelle multiplexer / demultiplexer described herein may include one or more force applicators operable to generate a monotonic refractive index across the free propagation region. For example, FIG3 shows a top view of a variation of a photonic integrated circuit 300 including a tunable echelle multiplexer / demultiplexer 301, which may be configured as described above with respect to the echelle multiplexer / demultiplexer 102 of FIG1A and FIG1B, except that the tunable echelle multiplexer / demultiplexer 301 includes a set of force applicators. In the variation shown in FIG3 , the echelle multiplexer / demultiplexer 301 includes force applicators positioned to apply forces on opposite sides of the free propagation region 110 (e.g., a first force applicator 302 positioned to apply a first force on a first side of the free propagation region 110 and a second force applicator 312 positioned to apply a second force on an opposite second side of the free propagation region 110). Additionally or alternatively, the echelle multiplexer / demultiplexer 301 may include multiple force applicators positioned to apply forces on a common side of the free propagation region 110. In still other variations, the echelle multiplexer / demultiplexer 301 may alternatively include a single force applicator positioned to apply a force on the first side of the free propagation region 110. Each force applicator is positioned and operable to apply a force to a respective portion of the slab waveguide forming the free propagation region 110, thereby generating a monotonic stress gradient across the free propagation region 110. For example, a first force applicator 302 can apply a first force to the waveguide layer 116 on a first side of the free propagation region 110, which can generate a first monotonic stress gradient across the free propagation region 110 that decreases along a first direction (e.g., along a first gradient direction 308 shown in FIG. 3 ). This, in turn, can generate a first monotonic refractive index gradient across the free propagation region 110 that increases along the first direction. Conversely, a second force applicator 312 can apply a second force to the waveguide layer 116 on a second side of the free propagation region 110, which can generate a second monotonic stress gradient across the free propagation region 110 that decreases along a second direction (e.g., along a second gradient direction 318 shown in FIG. 3 ). This, in turn, can generate a second monotonic refractive index gradient across the free propagation region 110 that increases along a second direction. In this manner, the first and second force applicators 302, 312 can be operated to selectively generate a monotonic refractive index gradient across the free propagation region 110 that increases or decreases along a given direction. Accordingly, the controller 306 can operate the first and second force applicators 302, 312 to correct for peak transmission wavelength shift relative to a target peak transmission wavelength, such as described in more detail with respect to the echelle multiplexer / demultiplexer 261 of FIG. 2F. Accordingly, the first force applicator 302 can be used to correct for peak transmission wavelength shift in a first direction (e.g., blue-shifting the peak transmission wavelength toward the target peak transmission wavelength), and the second force applicator 312 can be used to correct for peak transmission wavelength shift in a second direction opposite the first direction (e.g., red-shifting the peak transmission wavelength toward the target peak transmission wavelength). The echelle grating multiplexer / demultiplexer described herein may utilize any force applicator to apply force to the waveguide layer 116 of the photonic integrated circuit described herein. For example, in some variations, the force applicator may utilize a piezoelectric actuator to apply force to the waveguide layer 116. In other variations, the force applicator may utilize an electromagnetic actuator to apply force to the waveguide layer 116. It should be understood that a variety of devices may be used to controllably apply force to the photonic integrated circuit 300. Although the present disclosure is described above with respect to various exemplary embodiments and implementations, it should be understood that the various features, aspects, and functionalities described in one or more of the individual embodiments do not limit their applicability to the specific embodiment with which they are described, but rather may be applied alone or in various combinations to one or more of the embodiments of the present invention, whether or not such embodiments are described, and whether or not such features are presented as part of the described embodiments. Accordingly, the breadth and scope of the present invention should not be limited to any of the exemplary embodiments described above, but rather be defined by the claims set forth herein. 1B-1B: Line 2B-2B: Line 100: Photonic Integrated Circuit 102: Echelle Grating Multiplexer / Demultiplexer 104: Input Waveguides 106a-106c: Output Waveguide 108: Echelle Grating 110: Free Propagation Region 112: Substrate 114: First Cladding Layer 116: Waveguide Layer 118: Second Cladding Layer 120: Third Cladding Layer 200: Photonic Integrated Circuit 201: Tunable Echelle Grating Multiplexer / Demultiplexer 202: Heater 204: Light Source Unit 206: Controller 208: First Gradient Direction 220: Photonic Integrated Circuit 222: Doped Region 224: Through-hole 230: Photonic integrated circuit 232: Conductive material 240: Photonic integrated circuit 242: Conductive material 244: Portion 250: Photonic integrated circuit 251: Echelle grating multiplexer / demultiplexer 252a: First heater; Heater 252b: Second heater; Heater 261: Echelle grating multiplexer / demultiplexer 262a: First heater 262b: First heater 268: Second gradient direction 300: Photonic integrated circuit 301: Echelle grating multiplexer / demultiplexer 302: First force applicator 306: Controller 308: First gradient direction 312: Second force applicator 318: Second gradient direction The present disclosure may be more readily understood by referring to the following embodiments with reference to the accompanying drawings, in which like reference numerals designate like structural elements, and in which: [FIG. 1A] and [FIG. 1B] depict, respectively, a top view and a cross-sectional side view of a photonic integrated circuit including an echelle multiplexer / demultiplexer. [FIG. 2A] depicts a top view of a variant of a photonic integrated circuit including a tunable echelle multiplexer / demultiplexer having a heater. [FIG. 2B] through [FIG. 2D] depict cross-sectional side views of a variant of a photonic integrated circuit including a heater that can be used with the tunable echelle multiplexer / demultiplexer described herein. [FIG. 2E] and [FIG. 2F] show top views of a variant of a photonic integrated circuit including a tunable echelle multiplexer / demultiplexer having multiple heaters. [FIG. 3] depicts a top view of a variant of a photonic integrated circuit including a tunable echelle multiplexer / demultiplexer having a set of force applicators. Cross-hatching or shading is used in the drawings to generally clarify boundaries between adjacent elements and to aid in legibility. Accordingly, the presence or absence of cross-hatching or shading does not express or indicate a preference or requirement for any particular material, material properties, element proportions, element dimensions, commonality among similarly depicted elements, or any other characteristic, attribute, or property of any element depicted in the drawings. It should be understood that the proportions and sizes (relative or absolute) of the various features and elements (and sets and groups thereof) provided in the drawings, as well as the boundaries, distances, and positioning relationships presented therebetween, are merely intended to facilitate understanding of the various embodiments described herein, and accordingly, are not necessarily presented or drawn to scale, and are not intended to indicate a preference or requirement for the illustrated embodiments to the exclusion of embodiments described with reference thereto. 2B-2B: Line 104: Input waveguide 106a~106c: output waveguide 108: Echelle grating 110: Free propagation area 200: Photonic Integrated Circuits 201: Echelle grating multiplexer / demultiplexer 202: Heater 204: Light source unit 206: Controller 208: First gradient direction

Claims

1. A photonic integrated circuit comprising: a mid-stepped grating multiplexer / demultiplexer, comprising: a plurality of waveguides, comprising: a set of input waveguides; and a set of output waveguides; a mid-stepped grating; a free propagation region positioned between the mid-stepped grating and the plurality of waveguides; and a first heater positioned across the free propagation region to generate a first monotonic temperature gradient.

2. The photonic integrated circuit as described in claim 1, wherein: The first heater does not overlap with the free propagation region.

3. The photonic integrated circuit of claim 1 or claim 2, comprising: a planar waveguide forming the free propagation region, wherein: The first heater includes a first doped region of the planar waveguide.

4. The photonic integrated circuit as described in claim 1 or claim 2, wherein: The echelle grating multiplexer / demultiplexer includes a second heater positioned to generate a second monotonic temperature gradient across the free propagation region.

5. The photonic integrated circuit as described in claim 4, wherein: The first heater and the second heater are positioned on opposite sides of the free propagation region.

6. The photonic integrated circuit as described in claim 4, wherein: The first heater and the second heater are positioned on a common side of the free propagation region.

7. A photonic integrated circuit comprising: an echelle grating multiplexer / demultiplexer, comprising: a plurality of waveguides, comprising: a set of input waveguides; and a set of output waveguides; an echelle grating; and a free propagation region positioned between the echelle grating and the plurality of waveguides; a light source unit optically connected to the set of input waveguides; and a controller configured to: measure a light intensity using a power monitor optically connected to at least one of the set of output waveguides; and control the echelle grating multiplexer / demultiplexer based on the measured light intensity to generate a monotonic refractive index gradient across the free propagation region.

8. The photonic integrated circuit as described in claim 7, wherein: The echelle grating multiplexer / demultiplexer includes a first heater positioned on a first side of the free propagation region; and the controller is operatively connected to the first heater.

9. The photonic integrated circuit as described in claim 8, wherein: The echelle grating multiplexer / demultiplexer includes a second heater positioned on the first side of the free propagation region; and the controller is operatively connected to the second heater.

10. The photonic integrated circuit as described in claim 8, wherein: The echelle grating multiplexer / demultiplexer includes a second heater positioned on a second side of the free propagation region opposite the first side; and the controller is operatively connected to the second heater.

11. A photonic integrated circuit as claimed in any of claims 8 to 10, comprising: a planar waveguide forming the free propagation region, wherein: The first heater includes a first doped region of the planar waveguide.

12. The photonic integrated circuit as described in claim 7, wherein: The echelle grating multiplexer / demultiplexer includes a first force applicator positioned to selectively apply a force on a first side of the free propagation region; and the controller is operatively connected to the first force applicator.

13. The photonic integrated circuit as described in claim 8, wherein: The echelle grating multiplexer / demultiplexer includes a second force applicator positioned to selectively apply a force on a second side of the free propagation region; and the controller is operatively connected to the second force applicator.

14. A method of operating an echelle grating multiplexer / demultiplexer, the echelle grating multiplexer / demultiplexer comprising a set of input waveguides, a set of output waveguides, an echelle grating, and a free propagation region, the method comprising: generating a first monotonic refractive index gradient across the free propagation region during a first time period, the first monotonic refractive index gradient decreasing along a direction; and generating a second monotonic refractive index gradient across the free propagation region during a second time period, the second monotonic refractive index gradient increasing along the direction.

15. The method of claim 14, wherein generating the first monotonic refractive index gradient during the first time period comprises: generating a first monotonic temperature gradient across the free propagation region using a first heater positioned on a first side of the free propagation region, the first monotonic temperature gradient decreasing in that direction.

16. As in request item 15, wherein: A planar waveguide forms the free propagation region; and the first heater includes a doped region of the planar waveguide.

17. The method of claim 15 or claim 16, wherein generating the second monotonic refractive index gradient during the second time period comprises: generating a second monotonic temperature gradient across the free propagation region using a second heater positioned on a second side of the free propagation region, the second monotonic temperature gradient increasing in the direction thereof.

18. The method of claim 14, wherein generating the first monotonic refractive index gradient during the first time period comprises: applying a first force on a first side of the free propagation region to generate a first monotonic stress gradient across the free propagation region, the first monotonic stress gradient increasing in the direction thereof.

19. The method of claim 18, wherein generating the second monotonic refractive index gradient during the second time period comprises: applying a second force on a second side of the free propagation region to generate a second monotonic stress gradient across the free propagation region, the second monotonic stress gradient decreasing along the direction.

20. A method according to any one of claims 14 to 16, comprising: prior to the first time period, detecting a first peak transmission wavelength offset relative to a target peak transmission wavelength in a channel of the intermediate echelle grating multiplexer / demultiplexer; and prior to the second time period, detecting a second peak transmission wavelength offset relative to the target peak transmission wavelength in the same channel of the intermediate echelle grating multiplexer / demultiplexer, wherein: The first monotonic refractive index gradient is generated in response to the detection of the first peak transmission wavelength shift; and the second monotonic refractive index gradient is generated in response to the detection of the second peak transmission wavelength shift.

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