CN112068244B - Athermal arrayed waveguide grating - Google Patents
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- CN112068244B CN112068244B CN202010986581.6A CN202010986581A CN112068244B CN 112068244 B CN112068244 B CN 112068244B CN 202010986581 A CN202010986581 A CN 202010986581A CN 112068244 B CN112068244 B CN 112068244B
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- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
- G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
- G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
- G02B6/12007—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind forming wavelength selective elements, e.g. multiplexer, demultiplexer
- G02B6/12009—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind forming wavelength selective elements, e.g. multiplexer, demultiplexer comprising arrayed waveguide grating [AWG] devices, i.e. with a phased array of waveguides
- G02B6/12026—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind forming wavelength selective elements, e.g. multiplexer, demultiplexer comprising arrayed waveguide grating [AWG] devices, i.e. with a phased array of waveguides characterised by means for reducing the temperature dependence
Abstract
The invention provides a athermal arrayed waveguide grating, in the arrayed waveguide, the adjacent arrayed waveguide has different waveguide lengths, the corresponding adjacent arrayed waveguide has different waveguide widths, and the waveguide width of the arrayed waveguide is reduced along with the increase of the waveguide length of the arrayed waveguide; the design and the process are simple, the post-treatment process is not needed, the mass production and the commercialization can be adapted, and the performance is stable; the method has good process compatibility, and can be processed by adopting a standard CMOS (complementary metal oxide semiconductor); the wavelength division multiplexing of the multi-channel number can be realized. The athermal arrayed waveguide grating can realize insensitivity of central wavelength temperature on the premise of ensuring the optical path difference of adjacent arrayed waveguides to be fixed on the whole.
Description
Technical Field
The invention belongs to the field of optical communication and integrated optics, and relates to a athermal arrayed waveguide grating.
Background
With the development of the 5G Internet, people have more and more demand on data transmission. Optical communication has become the main method of network information transmission at present, but the demand for data transmission has increased very rapidly in recent years. In order to increase the network transmission bandwidth and reduce the network data transmission cost, the wavelength division multiplexing system has become a main solution for increasing the network bandwidth.
At present, there are two main methods for implementing wavelength division multiplexing: one is a wavelength division multiplexing device manufactured based on a geometric optical lens system, and the other is a wavelength division multiplexing device manufactured based on integrated optics. The wavelength division multiplexing device manufactured based on the geometric optical lens system is easy to realize, but has the defects of large device size, poor system stability and the like, and the wavelength division multiplexing device manufactured based on the integrated optics has the advantages of small device size, good performance stability and low cost, so that the wavelength division multiplexing device manufactured based on the integrated optics is more suitable for large-scale automatic production.
The wavelength division multiplexing device based on integrated optics is mainly Arrayed waveguide grating (Arrayed waveguide grating). In the wavelength division multiplexing device manufactured based on integrated optics, since the material itself has a thermo-optic effect and the refractive index of the material changes with the temperature, the AWG generally has temperature sensitivity, and the central wavelength shifts with the temperature, so that an additional temperature circuit control system is generally required to maintain the temperature stability of the device.
In modern communications, network operators prefer to build passive optical networks with low maintenance cost, so designing and manufacturing athermal passive AWG devices is a key point for implementing commercialization of integrated AWGs, although many athermal solutions have been proposed at this stage, mainly including: based on a metal compensation structure and based on a negative temperature coefficient material. However, although both methods have good temperature stability control effect, for SOI, SIN integrated optical processes, these two methods are not compatible with the conventional CMOS process, and cannot be produced in large quantities.
With the increasing demand of communication systems for miniaturization, low energy consumption and large bandwidth, the integrated optical technology will be one of the main directions for the development of future optical communication, and no heat on chip is a key technical point for the commercialization of passive AWG.
Therefore, it is necessary to provide an athermal arrayed waveguide grating.
Disclosure of Invention
In view of the above-mentioned shortcomings of the prior art, the present invention is directed to an athermal arrayed waveguide grating, which is used to solve the above-mentioned series of fabrication problems faced by the athermal arrayed waveguide in the prior art.
To achieve the above and other related objects, the present invention provides an athermal arrayed waveguide grating, comprising:
an input slab waveguide;
an output slab waveguide;
the array waveguide is connected with the input panel waveguide and the output panel waveguide, the adjacent array waveguides have different waveguide lengths, the corresponding adjacent array waveguides have different waveguide widths, and the waveguide widths of the array waveguides are reduced along with the increase of the waveguide lengths of the array waveguides;
an input waveguide connected to the input slab waveguide;
an output waveguide connected with the output slab waveguide.
Optionally, the arrayed waveguide comprises one of a rectangular waveguide, a ridge waveguide, and an ion-diffusion waveguide.
Optionally, the topography of the arrayed waveguide comprises a saddle or a three-segment topography.
Optionally, when the shape of the arrayed waveguide adopts a saddle shape, the arrayed waveguide comprises one or a combination of a single-mode rectangular waveguide in the curved transmission part and a ridge-shaped straight waveguide in the straight transmission part.
Optionally, further comprising a tapered waveguide comprising one or a combination of between the input slab waveguide and the arrayed waveguide and between the output slab waveguide and the arrayed waveguide.
Optionally, in the athermal arrayed waveguide grating, the material of the waveguide core layer includes one of silicon, silicon nitride, silicon oxynitride, and silicon dioxide, the material of the waveguide cladding layer includes one of silicon dioxide, silicon nitride, silicon oxynitride, and air, and the refractive index of the waveguide core layer is greater than that of the waveguide cladding layer.
Optionally, the input waveguide comprises M ≧ 1.
Optionally, the waveguide width of the arrayed waveguide ranges from 0.2 μm to 10 μm.
Optionally, the thermo-optic coefficient of the effective refractive index of the arrayed waveguide increases with increasing waveguide width; the effective refractive index of the arrayed waveguide increases with increasing waveguide width.
Optionally, the effective refractive index of the arrayed waveguide is linear with temperature, and the athermal arrayed waveguide grating satisfies the following conditions:
wherein n isi+1Is the effective refractive index of the (i + 1) th arrayed waveguide; n isiIs the effective refractive index of the ith said arrayed waveguide; l isi+1The waveguide length of the (i + 1) th arrayed waveguide; l isiThe waveguide length of the ith array waveguide; n issIs the effective refractive index of the output slab waveguide; daThe distance between adjacent arrayed waveguides; theta is an included angle between the central waveguide and the output waveguide; m is a diffraction order; λ is the center wavelength; t is the temperature; t is0Is the initial temperature; k is a radical ofi+1The thermo-optic coefficient of the (i + 1) th arrayed waveguide; k is a radical ofiThe thermo-optic coefficient of the ith arrayed waveguide.
As described above, in the athermal arrayed waveguide grating of the present invention, the adjacent arrayed waveguides have different waveguide lengths, the corresponding adjacent arrayed waveguides have different waveguide widths, and the waveguide width of the arrayed waveguide decreases with the increase of the waveguide length of the arrayed waveguide, and by changing the waveguide width of the arrayed waveguide, athermalization of the arrayed waveguide grating is achieved, and good temperature stability can be maintained without external energy supply, so that the athermal arrayed waveguide grating is a passive device, and the system maintenance cost can be greatly reduced; the design and the process are simple, the post-treatment process is not needed, the mass production and the commercialization can be adapted, and the performance is stable; the method has good process compatibility, and can be processed by adopting a standard CMOS (complementary metal oxide semiconductor); the wavelength division multiplexing with multiple channels can be realized, and the wavelength division multiplexing comprises coarse wavelength division and dense wavelength division. The athermal arrayed waveguide grating can realize insensitivity of central wavelength temperature on the premise of ensuring the optical path difference of adjacent arrayed waveguides to be fixed on the whole.
Drawings
Fig. 1 is a schematic structural diagram of an athermal arrayed waveguide grating with a three-segment arrayed waveguide according to an embodiment.
Fig. 2 is a schematic cross-sectional view of a rectangular waveguide in the example.
Fig. 3 is a graph showing the effective refractive index of arrayed waveguides having different waveguide widths as a function of temperature in the example.
FIG. 4 is a graph showing the relationship between the effective refractive index of arrayed waveguides having different waveguide widths and the thermo-optic coefficient of the effective refractive index in the examples.
Fig. 5 is a schematic structural diagram of an athermal arrayed waveguide grating with saddle-shaped arrayed waveguides in an embodiment.
Description of the element reference numerals
101 input waveguide
102 input slab waveguide
103 arrayed waveguide
104 output slab waveguide
105 output waveguide
201 waveguide substrate
202 rectangular waveguide core layer
203 rectangular waveguide cladding
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention.
As in the detailed description of the embodiments of the present invention, the cross-sectional views illustrating the device structures are not partially enlarged in general scale for convenience of illustration, and the schematic views are only examples, which should not limit the scope of the present invention. In addition, the three-dimensional dimensions of length, width and depth should be included in the actual fabrication.
For convenience in description, spatial relational terms such as "below," "beneath," "below," "under," "over," "upper," and the like may be used herein to describe one element or feature's relationship to another element or feature as illustrated in the figures. It will be understood that these terms of spatial relationship are intended to encompass other orientations of the device in use or operation in addition to the orientation depicted in the figures. Further, when a layer is referred to as being "between" two layers, it can be the only layer between the two layers, or one or more intervening layers may also be present. As used herein, "between … …" is meant to include both endpoints.
In the context of this application, a structure described as having a first feature "on" a second feature may include embodiments in which the first and second features are formed in direct contact, and may also include embodiments in which additional features are formed in between the first and second features, such that the first and second features may not be in direct contact.
It should be noted that the drawings provided in the present embodiment are only for illustrating the basic idea of the present invention, and the drawings only show the components related to the present invention rather than being drawn according to the number, shape and size of the components in actual implementation, and the type, quantity and proportion of each component in actual implementation may be changed freely, and the layout of the components may be more complicated.
It should be noted that the waveguide width in the present invention refers to the waveguide core width as understood by those skilled in the art.
Referring to fig. 1, the present embodiment provides an athermal arrayed waveguide grating, which includes an input waveguide 101, an input slab waveguide 102, an arrayed waveguide 103, an output slab waveguide 104, and an output waveguide 105; the input waveguide 101 is connected to the input slab waveguide 102, the output waveguide 105 is connected to the output slab waveguide 104, the arrayed waveguide 103 is connected to the input slab waveguide 102 and the output slab waveguide 104, adjacent arrayed waveguides 103 have different waveguide lengths L, corresponding adjacent arrayed waveguides 103 have different waveguide widths W, and the waveguide widths W decrease with the increase of the waveguide lengths L of the arrayed waveguides 103.
The athermal arrayed waveguide grating of the embodiment realizes athermalization of the arrayed waveguide grating by changing the waveguide width W of the arrayed waveguide 103, can maintain good temperature stability without external energy supply, is a passive device, and can greatly reduce the system maintenance cost; the design and the process are simple, the post-treatment process is not needed, the mass production and the commercialization can be adapted, and the performance is stable; the method has good process compatibility, and can be processed by adopting a standard CMOS (complementary metal oxide semiconductor); the wavelength division multiplexing with multiple channels can be realized, and the wavelength division multiplexing comprises coarse wavelength division and dense wavelength division. The athermal arrayed waveguide grating of this embodiment can realize insensitivity of center wavelength temperature on the premise of ensuring that the optical path difference between adjacent arrayed waveguides 103 is fixed as a whole.
Specifically, the key to the implementation of the wavelength division multiplexing by the arrayed waveguide grating is that in the arrayed waveguide, the difference between the waveguide lengths L of the adjacent arrayed waveguides 103 is a fixed value Δ L, wherein the value of Δ L determines the diffraction order m and the central wavelength λ of the arrayed waveguide grating. In a conventional arrayed waveguide grating, the waveguide width W and the waveguide height corresponding to an arrayed waveguide are the same, so that when the ambient temperature changes, due to the thermo-optic effect, the effective refractive index N of the arrayed waveguide changes with the change of the temperature T, and if the change amount of the effective refractive index N of the arrayed waveguide is Δ N, in the arrayed waveguide, the optical path difference of adjacent arrayed waveguides generates a drift amount of Δ N × Δ L, and the central wavelength λ of the corresponding arrayed waveguide grating also shifts, thereby causing the temperature sensitivity of the device.
Suppose that the waveguide lengths of the adjacent arrayed waveguides are respectively Li+1，LiThe effective refractive index of the corresponding array waveguide is ni+1，niThen the optical path difference between adjacent arrayed waveguides can be expressed as: n isi+1×Li+1－ni×Li. When the effective refractive index n of the array waveguide generates delta n along with the temperature Ti+1，ΔniIf the adjacent arrayed waveguide width W is the same as the waveguide height, then ni+1＝ni，Δn＝Δni+1＝ΔniThen n isi+1+Δn＝ni+ Δ n; and the length difference of the array waveguide is a fixed value delta L-Li+1－LiThen the optical path difference drift amount of the adjacent arrayed waveguide is Δ n × Δ L. Therefore, in the conventional arrayed waveguide grating, the optical path drift is generated due to the change of the temperature T, and the long waveguide generates more optical path offset with the same change of the effective refractive index.
As an example, the thermo-optic coefficient of the effective refractive index of the arrayed waveguide 103 increases with increasing waveguide width; the effective refractive index of the arrayed waveguide 103 increases with increasing waveguide width.
As an example, the effective refractive index of the arrayed waveguide 103 varies linearly with temperature, and the athermal arrayed waveguide grating satisfies:
wherein n isi+1Is the effective refractive index of the (i + 1) th arrayed waveguide 103; n isiIs the effective refractive index of the ith said arrayed waveguide 103; l isi+1The waveguide length of the (i + 1) th arrayed waveguide 103; l isiThe waveguide length of the ith array waveguide 103; n issIs the effective refractive index of the output slab waveguide; daThe pitch of the adjacent arrayed waveguides 103; theta is an included angle between the central waveguide and the output waveguide; m is a diffraction order; λ is the center wavelength; t is the temperature; t is0Is the initial temperature; k is a radical ofi+1Is the thermo-optic coefficient of the (i + 1) th arrayed waveguide 103; k is a radical ofiIs the thermo-optic coefficient of the ith arrayed waveguide 103.
Specifically, according to the waveguide theory, the effective refractive index n of the waveguide is determined by the refractive indexes of the cladding and core layer materials, and when the waveguide size is small, the cladding material contributes greatly to the effective refractive index of the waveguide; when the waveguide size is large, the contribution of the core material to the waveguide effective index is large. Usually the effective refractive index of different materials differs in thermo-optic coefficient, e.g. 1.8 x 10 for silicon-4Per DEG C, the thermo-optic coefficient of silicon nitride is 0.38X 10-4Per DEG C, the thermo-optic coefficient of silicon dioxide is 0.1X 10-4V. C. In the embodiment shown in fig. 2, the arrayed waveguide may be a rectangular waveguide, but is not limited thereto, and may also be a ridge waveguide or an ion-diffusion waveguide, for example, without being limited thereto. On the waveguide substrate 201, silicon nitride is used for the rectangular waveguide core layer 202, and silicon dioxide is used for the rectangular waveguide cladding layer 203, but the selection of the material of the waveguide core layer and the material of the waveguide cladding layer is not limited thereto and is not limited herein. For example, the material of the waveguide core layer may include one of silicon, silicon nitride, silicon oxynitride and silicon dioxide, the material of the waveguide cladding layer may include one of silicon dioxide, silicon nitride, silicon oxynitride and air, and the refractive index of the waveguide core layer is greater than that of the waveguide cladding layer.
Referring to fig. 3, the effective refractive index of the arrayed waveguide having different waveguide widths W is analyzed with respect to temperature change by using silicon nitride as the rectangular waveguide core layer 202 and silicon dioxide as the rectangular waveguide cladding layer 203.
The core material contributes more to the waveguide effective refractive index as the waveguide width W increases, and since the core material silicon nitride has a larger thermo-optic coefficient than silicon dioxide, the thermo-optic coefficient that results in the effective refractive index of the arrayed waveguide 103 also increases as the waveguide width W increases.
As can be seen from fig. 3, the effective refractive index of the arrayed waveguide 103 is linear with temperature, that is, the following conditions are satisfied:
thus, it is possible to obtain:
therefore, by appropriately reducing the width of the (i + 1) th arrayed waveguide 103 by utilizing the relation between the thermo-optic coefficient and the waveguide width WWi+1Or increasing the width W of the ith arrayed waveguide 103iThe effective refractive index of the long waveguide can be changed by an amount Δ ni+1Effective refractive index change Δ n smaller than short waveguideiTherefore, on the premise of ensuring that the optical path difference of the adjacent arrayed waveguides 103 is fixed on the whole, the athermal arrayed waveguide grating with insensitive central wavelength lambda temperature can be realized. In this embodiment, in the non-uniform arrayed waveguide 103, the waveguide widths W of the adjacent arrayed waveguides 103 are different, the shorter the inside arrayed waveguide 103 has the wider waveguide width W, and the longer the length of the outside arrayed waveguide 103 is, the narrower the waveguide width W of the corresponding arrayed waveguide 103 is.
Referring to fig. 1, including 1, 2, 3-N arrayed waveguides 103, the athermal arrayed waveguide grating satisfies the grating equation:
ni+1Li+1-niLi+nsdasinθ＝mλ (1)；
the temperature T is differentiated according to equation 1:
here θ ≈ 0, so combining (1) and (2) can yield the system of equations:
because:
substituting the above equation set (3) can obtain:
solving equation set (4) yields:
then, the following can be obtained:
wherein, optionally, one of the arrayed waveguides 103 is used as a reference, i.e., a known n is providedi、LiAnd kiIf n is selectable1、L1And k1Then according to ni+1And k isi+1And by combining the experimental or simulated relationship between the effective refractive index of the arrayed waveguide 103 having different waveguide widths and the thermo-optic coefficient of the effective refractive index as shown in fig. 4, the waveguide width W of the (i + 1) th arrayed waveguide 103 can be obtainedi+1。
By way of example, the topography of the arrayed waveguide 103 includes a saddle or a three-segment topography.
Specifically, referring to fig. 1, fig. 1 only shows the waveguide core layer in the athermal arrayed waveguide grating for understanding, in the embodiment, the arrayed waveguide 103 adopts a three-segment type morphology, that is, three segments a, B and c, but the morphology of the arrayed waveguide 103 may also include a saddle shape, as shown in fig. 5, in which the saddle-shaped arrayed waveguide 103 includes a curved transmission portion a and a linear transmission portion B.
As an example, when the shape of the arrayed waveguide 103 is saddle-shaped, the arrayed waveguide 103 includes one or a combination of a single-mode rectangular waveguide in the curved transmission section a and a ridge-shaped straight waveguide in the straight transmission section B to reduce transmission loss and form a transmission loss optimization structure, and the type of the transmission loss optimization structure is not limited herein.
As an example, a tapered waveguide is further included, and the tapered waveguide includes one or a combination of a position between the input slab waveguide 102 and the arrayed waveguide 103 and a position between the output slab waveguide 104 and the arrayed waveguide 103, so as to serve as a coupling efficiency optimization structure through the tapered waveguide, and the kind of the coupling efficiency optimization structure is not limited herein.
By way of example, the input waveguide 101 may include M ≧ 1, and the specific number may be selected as desired without undue limitation.
As an example, in the arrayed waveguide, the range of the waveguide width W of the arrayed waveguide 103 includes 0.2 μm to 10 μm.
Specifically, referring to fig. 4, the waveguide width W of the arrayed waveguide 103 may be any value within a range of 0.5 μm, 0.7 μm, 0.9 μm, 1.0 μm, 1.4 μm, 2 μm, 5 μm, 10 μm, or the like.
In summary, in the athermal arrayed waveguide grating of the present invention, the adjacent arrayed waveguides have different waveguide lengths, the corresponding adjacent arrayed waveguides have different waveguide widths, and the waveguide width of the arrayed waveguide decreases with the increase of the waveguide length of the arrayed waveguide, and by changing the waveguide width of the arrayed waveguide, athermalization of the arrayed waveguide grating is achieved, and good temperature stability can be maintained without external energy supply, so that the athermal arrayed waveguide grating is a passive device, and the system maintenance cost can be greatly reduced; the design and the process are simple, the post-treatment process is not needed, the mass production and the commercialization can be adapted, and the performance is stable; the method has good process compatibility, and can be processed by adopting a standard CMOS (complementary metal oxide semiconductor); the wavelength division multiplexing with multiple channels can be realized, and the wavelength division multiplexing comprises coarse wavelength division and dense wavelength division. The athermal arrayed waveguide grating can realize insensitivity of central wavelength temperature on the premise of ensuring the optical path difference of adjacent arrayed waveguides to be fixed on the whole.
The foregoing embodiments are merely illustrative of the principles and utilities of the present invention and are not intended to limit the invention. Those skilled in the art can modify or change the above-described embodiments without departing from the spirit and scope of the present invention. Accordingly, it is intended that all equivalent modifications or changes which can be made by those skilled in the art without departing from the spirit and technical spirit of the present invention be covered by the claims of the present invention.
Claims (10)
1. An athermal arrayed waveguide grating, comprising:
an input slab waveguide;
an output slab waveguide;
the array waveguide is connected with the input panel waveguide and the output panel waveguide, the adjacent array waveguides have different waveguide lengths, the corresponding adjacent array waveguides have different waveguide widths, and the waveguide widths of the array waveguides are reduced along with the increase of the waveguide lengths of the array waveguides;
an input waveguide connected to the input slab waveguide;
an output waveguide connected with the output slab waveguide.
2. The athermal arrayed waveguide grating of claim 1, wherein: the arrayed waveguide includes one of a rectangular waveguide, a ridge waveguide, and an ion-diffusion waveguide.
3. The athermal arrayed waveguide grating of claim 1, wherein: the shape of the arrayed waveguide comprises a saddle shape or a three-section shape.
4. The athermal arrayed waveguide grating of claim 3, wherein: when the shape of the arrayed waveguide adopts a saddle shape, the arrayed waveguide comprises one or a combination of a single-mode rectangular waveguide adopted in a curved transmission part and a ridge-shaped straight waveguide adopted in a straight transmission part.
5. The athermal arrayed waveguide grating of claim 1, wherein: further comprising a tapered waveguide comprising one or a combination of between the input slab waveguide and the arrayed waveguide and between the output slab waveguide and the arrayed waveguide.
6. The athermal arrayed waveguide grating of claim 1, wherein: in the athermal arrayed waveguide grating, the waveguide core layer is made of one of silicon, silicon nitride, silicon oxynitride and silicon dioxide, the waveguide cladding layer is made of one of silicon dioxide, silicon nitride, silicon oxynitride and air, and the refractive index of the waveguide core layer is larger than that of the waveguide cladding layer.
7. The athermal arrayed waveguide grating of claim 1, wherein: the input waveguide comprises M which is more than or equal to 1.
8. The athermal arrayed waveguide grating of claim 1, wherein: the waveguide width of the arrayed waveguide ranges from 0.2 μm to 10 μm.
9. The athermal arrayed waveguide grating of claim 1, wherein: the thermo-optic coefficient of the effective refractive index of the arrayed waveguide increases with the increase of the waveguide width; the effective refractive index of the arrayed waveguide increases with increasing waveguide width.
10. The athermal arrayed waveguide grating of claim 1, wherein the effective refractive index of the arrayed waveguide varies linearly with temperature, and the athermal arrayed waveguide grating satisfies:
wherein n isi+1Is the effective refractive index of the (i + 1) th arrayed waveguide; n isiIs the effective refractive index of the ith said arrayed waveguide; l isi+1The waveguide length of the (i + 1) th arrayed waveguide; l isiThe waveguide length of the ith array waveguide; n issIs the effective refractive index of the output slab waveguide; daThe distance between adjacent arrayed waveguides; theta is an included angle between the central waveguide and the output waveguide; m is a diffraction order; λ is the center wavelength;t is the temperature; t is0Is the initial temperature; k is a radical ofi+1The thermo-optic coefficient of the (i + 1) th arrayed waveguide; k is a radical ofiThe thermo-optic coefficient of the ith arrayed waveguide.
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