JP4069102B2 - Waveguide type optical multiplexer / demultiplexer - Google Patents

Description

  The present invention relates to a waveguide-type optical multiplexing / demultiplexing circuit, and more particularly, to a waveguide-type optical multiplexing / demultiplexing circuit using a holographic wave transmission medium that transmits waves holographically by multiple scattering according to a two-dimensional refractive index distribution. The present invention relates to a branching circuit.

  Currently, in order to expand communication capacity, development of an optical wavelength division multiplexing (WDM) transmission system that transmits optical signals of different wavelengths through a single optical fiber is in progress. In order to apply a WDM transmission system conventionally used as a trunk transmission system to an access transmission system, cost reduction is an issue. Therefore, a CWDM (Coarse WDM) transmission system has been developed in which the wavelength interval is increased to reduce the cost of optical components. As one of the optical components of the WDM transmission system, light that combines optical signals of a plurality of wavelengths on the transmitter side and demultiplexes a plurality of optical signals from one optical fiber to different ports on the receiver side. A wavelength multiplexing / demultiplexing circuit is used.

  FIG. 1 shows a configuration of a conventional arrayed waveguide diffraction grating type optical multiplexing / demultiplexing circuit. The arrayed waveguide grating optical multiplexing / demultiplexing circuit includes an input waveguide 101, a first slab waveguide 102, an arrayed waveguide 103, a second slab waveguide 104, and an output waveguide on a substrate 100. The waveguide 105 is connected and disposed in order (for example, see Non-Patent Document 1). The optical signal guided to the input waveguide 101 has its optical path changed in accordance with the wavelength in the first slab waveguide 102 and branched to the corresponding arrayed waveguide 103. The optical signals that have passed through the arrayed waveguide 103 are combined again by the second slab waveguide 104 and guided to the output waveguide 105.

  Here, the optical field pattern projected onto the array waveguide end of the first slab waveguide 102 is copied to the array waveguide end of the second slab waveguide 104. The arrayed waveguide 103 is designed so that the optical path lengths of adjacent optical waveguides differ by ΔL, and the field has an inclination depending on the wavelength of the input light. Due to this inclination, at the output waveguide end of the second slab waveguide 104, the position where the optical field is focused changes for each wavelength, and wavelength demultiplexing becomes possible.

  On the other hand, if an optical signal having a plurality of different wavelengths is input to the output waveguide 105, an optical signal that is optical wavelength division multiplexed is output from the input waveguide 101. That is, wavelength multiplexing can be performed by the arrayed waveguide grating optical multiplexing / demultiplexing circuit.

K. Okamoto, “Fundamentals of Optical Waveguides”, Academic Press, 2000

  However, in the CWDM transmission system, since the channel wavelength interval is as wide as 20 nm, it is necessary to make the optical path length difference ΔL as extremely small as 6.4 μm. Therefore, from the viewpoint of circuit layout, the arrayed waveguide 103 is divided into a plurality of blocks, and a small optical path length difference is realized by the difference of ΔL for each block. In FIG. 1, it is divided into two blocks of the arrayed waveguides 103a and 103c and the arrayed waveguide 103b, and two ΔL having a small difference are combined. Such a circuit layout has a problem that the substrate size becomes large.

  The present invention has been made in view of such problems, and an object thereof is to provide a waveguide-type optical multiplexing / demultiplexing circuit having an arrayed waveguide having a small optical path length difference.

According to the first aspect of the present invention, each of the first slab waveguide that branches the optical signals received from the plurality of input waveguides and the plurality of waveguides that are sequentially increased by a predetermined waveguide length difference, An arrayed waveguide for inputting each optical signal branched by one slab waveguide; a second slab waveguide for guiding each signal light output from the arrayed waveguide to each of a plurality of output waveguides; A waveguide-type optical multiplexing / demultiplexing circuit including a wave transmission medium in which a spatial refractive index distribution is formed in a core layer of the arrayed waveguide in a part of the arrayed waveguide, and the spatial refractive index The distribution is such that when input light incident on the wave transmission medium from the array waveguide propagates along the light propagation direction, the input field of the input light is converted into an output field of output light, and the array one or more input and output to be output to the waveguide To propagate each set of field is determined by the refractive index with the core of each pixel defined by the mesh, the refractive index of the core of each of the pixels in the light propagation direction of the array waveguide coordinate z, in the cross section X of the location when the coordinate x in the direction perpendicular to the propagation direction of the light in the core layer (z, x), the phase of the forward propagation of the field of the input field, before Symbol output It is determined by repeatedly calculating using the refractive index of the core of each of the pixels as a variable so that the phase difference between the phase propagated back to the phase conjugate of the field is equal to or less than a predetermined error. The calculation is based on the refractive index distribution {n _q−1 } obtained by the (q−1) th calculation, and the jth input field ψ ^j (x) and the output field of the set of input / output fields. For the field φ ^j (x), the forward propagation field ψ ^j (z, x, {n _q-1 }) and the back propagation field φ ^j (z, x, {n _q-1 }) The refractive index n _q (z, x) at each location (z, x) is
n _q (z, x) = n _q-1 (z, x) −αΣ _j Im [φ ^j (z, x, {n _q-1 }) ^* · ψ ^j (z, x, {n _q-1 })]
Where the symbol “*” is a complex conjugate, the symbol “•” is an inner product operation, Im [] is the imaginary component of the field inner product operation result in [], and α is the convergence of the calculation Σ _j represents the sum of j, and the field of the input light has a phase change in multiple stages by the refractive index distribution of the cross section X along the propagation direction of the arrayed waveguide. By receiving, it propagates, changing the shape of the field of light by the interference phenomenon by the multiple scattering of the propagation waves which generate | occur | produce in the said wave transmission medium.

In the above-described waveguide-type optical multiplexing / demultiplexing circuit, the minimum dimension of the pixel is 0.2 μm or more. Furthermore, the waveguide is formed of a silica-based glass optical waveguide on a silicon substrate.

As described above, according to the present invention, since the wave transmission medium in which the spatial refractive index distribution is formed in the core layer of the waveguide is disposed in a part of the arrayed waveguide, the effective efficiency of the arrayed waveguide is increased. The refractive index can be changed, and an arrayed waveguide having a small optical path length difference can be realized.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. In the arrayed waveguide diffraction grating type optical multiplexing / demultiplexing circuit of this embodiment, a plurality of scattering points having a refractive index lower than that of the waveguide are arranged in a part of the arrayed waveguide. A holographic wave transmission medium defined by a plurality of scattering points transmits waves to the holographic wave by multiple scattering according to a two-dimensional refractive index distribution. Since the effective refractive index of the arrayed waveguide can be changed by the holographic wave transmission medium, the optical path length of the optical signal propagating through the arrayed waveguide can be shortened.

  First, the basic concept of the wave transmission medium used in the present invention will be described. Here, since it is applied to an optical circuit, the “wave” propagating in the wave transmission medium is “light”. The theory relating to the wave transmission medium specifies the characteristics of the medium based on a general wave equation, and can also hold in principle in general waves. In order to input a coherent light pattern and output a desired light pattern, the wave transmission medium has a phase difference between forward propagation light and back propagation light propagating in the wave transmission medium. The refractive index distribution is determined so as to be small even at the location. A desired light pattern is output by repeatedly repeating holographic control at a local level according to the refractive index distribution.

The basic structure of the wave transmission medium according to the present embodiment will be described with reference to FIG. As shown in FIG. 2A, an optical circuit design region 1-1 configured by a wave transmission medium exists in the optical circuit board 1. One end surface of the optical circuit is an incident surface 2-1 on which the input light 3-1 is incident. The input light 3-1 propagates through the optical circuit having a spatial refractive index distribution constituted by the wave transmission medium while being subjected to multiple scattering, and is output as the output light 3-2 from the exit surface 2-2 which is the other end face. Is output. Coordinate z in FIG. 2 (a), the coordinates (z = 0 is the incident surface, z = z _e is output surface) of the propagation direction of the light is, the coordinate x is at a transverse coordinates for the light propagation direction is there. In this embodiment, it is assumed that the wave transmission medium is made of a dielectric, and the spatial refractive index distribution is based on the theory described below for the local refractive index of the dielectric constituting the wave transmission medium. This is realized by setting based on this.

  The “field” (input field) formed by the input light 3-1 is modulated according to the spatial distribution of the refractive index of the wave transmission medium constituting the optical circuit, and the “field” formed by the output light 3-2 "(Output field). In other words, the wave transmission medium of the present invention is (electromagnetic) field conversion means for correlating the input field and the output field according to the spatial refractive index distribution. For these input field and output field, the light field in the cross section (cross section along the x axis in the figure) perpendicular to the propagation direction in the optical circuit (z axis direction in the figure) This is referred to as a (forward) propagation image (propagation field or propagation light) in z) (see FIG. 2B).

  Here, “field” generally means an electromagnetic field (electromagnetic field) or a vector potential field of an electromagnetic field. The control of the electromagnetic field in this embodiment corresponds to changing the spatial refractive index distribution provided in the optical circuit, that is, the dielectric constant distribution. Although the dielectric constant is given as a tensor, since the transition between the polarization states is usually not so large, it is an approximation that may be a scalar wave approximation for only one component of the electromagnetic field. Therefore, in this specification, the electromagnetic field is treated as a complex scalar wave. Since the “state” of light includes an energy state (wavelength) and a polarization state, when the “field” is used to express the state of light, the light wavelength and polarization state are also included. Will get.

  In general, in an optical circuit that does not cause amplification or attenuation of propagating light, when the spatial distribution of the refractive index is determined, an image (input field) of the input light 3-1 other than the focal point is an image of the output light 3-2. (Output field) is determined uniquely. Such a field of light traveling from the exit surface 2-2 side to the entrance surface 2-1 side is referred to as a back propagation image (a back propagation field or a back propagation light) (see FIG. 2C). Such a back propagation image can be defined for each place in the optical circuit. That is, when a light field at an arbitrary place in the optical circuit is considered, if the place is considered as an emission point of a virtual “input light”, the image of the output light 3-2 can be obtained in the same manner as described above. The back-propagation image at that location can be considered. In this way, a back propagation image can be defined for each location in the optical circuit.

  In particular, in a single optical circuit, when the outgoing field is a propagation field of the incident field, the propagation field and the reverse propagation field coincide with each other at any point of the optical circuit. The field is generally a function over the entire target space, but the term “incident field” or “exit field” means a cross section of the field on the entrance surface or exit surface. In addition, even in the case of “field distribution”, when a discussion is made regarding a specific cross section, it means a cross section of the field with respect to that cross section.

In order to explain the method of determining the refractive index distribution, it is better to use symbols, so the following symbols are used to represent each quantity. Note that the target light (field) is not limited to light in a single state. Therefore, an index j is assigned to light in each state so that light in which a plurality of states are superimposed can be targeted. In general.
Ψ ^j (x): j-th incident field (complex vector value function, defined by intensity distribution and phase distribution set on the incident surface, and wavelength and polarization)
Φ ^j (x): j-th outgoing field (complex vector value function, defined by intensity distribution and phase distribution, wavelength and polarization set on the outgoing face)
Note that ψ ^j (x) and φ ^j (x) have the same total light intensity (or a negligible loss) unless intensity amplification, wavelength conversion, and polarization conversion are performed in the circuit. Their wavelength and polarization are the same.
{Ψ ^j (x), φ ^j (x)}: Input / output pair (a set of input / output fields)
{Ψ ^j (x), φ ^j (x)} is defined by the intensity distribution and the phase distribution, the wavelength and the polarization at the entrance surface and the exit surface.
{N _q }: Refractive index distribution (a set of values for the entire optical circuit design area)
Since one field of light is determined when one refractive index distribution is given to a given incident field and outgoing field, it is necessary to consider a field for the entire refractive index distribution given by the qth iterative operation. Therefore, the entire refractive index distribution may be expressed as n _q (x, z) with (x, z) as an indefinite variable, but the refractive index value n _q (x, z) at the location (x, z) For distinction, {n _q } is used for the entire refractive index distribution.
N _core : A symbol indicating a high refractive index value with respect to the surrounding refractive index, such as a core portion in an optical waveguide.
N _clad : a symbol indicating a low refractive index value with respect to n _core , such as a clad portion in an optical waveguide.
Ψ ^j (z, x, {n _q }): field at location (x, z) when the ^jth incident field ψ ^j (x) is propagated through the refractive index profile {n _q } to z The value of the.
Φ ^j (z, x, {n _q }): at the location (x, z) when the j-th outgoing field φ ^j (x) is propagated back to z in the refractive index profile {n _q } The field value.

In the present embodiment, the refractive index distribution, all j for _{^{ψ j (z e, x,}} {n q}) is given = φ ^j (x), or the so close state {n _q}.

  “Input port” and “output port” are “areas” where the fields at the entrance end face and the exit end face are concentrated. For example, by connecting an optical fiber to the part, the optical intensity can be propagated to the fiber. It is an area. Here, since the field intensity distribution and phase distribution can be designed to be different for the j-th and k-th ones, a plurality of ports can be provided on the entrance end face and the exit end face. Furthermore, when considering a pair of an incident field and an outgoing field, the phase generated by the propagation between them differs depending on the frequency of light. Therefore, for light having different frequencies (that is, light having different wavelengths), the field shape including the phase is included. Regardless of whether they are the same or orthogonal, they can be configured as different ports.

Here, the electromagnetic field is a field of a real vector value, and has a wavelength and a polarization state as parameters, but the value of the component is displayed as a complex number that can be easily handled in a general mathematical manner, and represents the solution of the electromagnetic wave. In the following calculation, it is assumed that the strength of the entire field is normalized to 1. As shown in FIGS. 2B and 2C, with respect to the jth incident field ψ ^j (x) and the output field φ ^j (x), the propagation field and the back propagation field are set at the respective locations. As complex vector value functions, they are expressed as ψ ^j (z, x, {n}) and φ ^j (z, x, {n}). Since the values of these functions vary depending on the refractive index distribution {n}, the refractive index distribution {n} is a parameter. According to the definition of the symbols, ψ ^j (x) = ψ ^j (0, x, {n}) and φ ^j (x) = φ ^j (z _e , x, {n}). The values of these functions can be easily calculated by a known method such as a beam propagation method, given an incident field ψ ^j (x), an outgoing field φ ^j (x), and a refractive index distribution {n}. it can.

In the following, a general algorithm for determining the spatial refractive index distribution will be described. FIG. 3 shows a calculation procedure for determining the spatial refractive index distribution of the wave transmission medium. Since this calculation is repeatedly executed, the number of repetitions is represented by q, and the state of the qth calculation when the calculation is executed up to the (q−1) th is shown. Based on the refractive index distribution {n _q-1 } obtained by the (q-1) th calculation, the propagation field and the ^jth incident field ψ ^j (x) and the outgoing field φ ^j (x) The back propagation field is obtained by numerical calculation, and the results are expressed as ψ ^j (z, x, {n _q-1 }) and φ ^j (z, x, {n _q-1 }), respectively (step S220). ).

Based on these results, the refractive index n _q (z, x) at each location (z, x) is obtained by the following equation (step S240).
n _q (z, x) = n _q-1 (z, x) −αΣ _j Im [φ ^j (z, x, {n _q-1 }) ^* · ψ ^j (z, x, {n _q-1 })] (1)
Here, the symbol “·” in the second term on the right side means an inner product operation, and Im [] means an imaginary number component of the field inner product operation result in []. The symbol “*” is a complex conjugate. The coefficient α is a value obtained by dividing a value equal to or less than a fraction of n _q (z, x) by the number of field pairs, and is a small positive value. Σ _j means that the sum is taken for the index j.

Step S220 and S240 and repeats the value in the exit plane of the propagation field _{^{ψ j (z e, x,}} {n}) the absolute value of the difference between the exit field φ ^j (x) is than desired error d _j When it becomes smaller (step S230: YES), the calculation ends.

In the above calculation, the initial value {n ₀ } of the refractive index distribution may be set appropriately. However, if this initial value {n ₀ } is close to the expected refractive index distribution, the calculation converges faster accordingly ( Step S200). When calculating φ ^j (z, x, {n _q-1 }) and ψ ^j (z, x, {n _q-1 }) for each j, , J (that is, φ ^j (z, x, {n _q-1 }) and ψ ^j (z, x, {n _q-1 })). Thus, calculation efficiency can be improved (step S220). If the computer is configured with a relatively small memory, an appropriate j is selected for each q in the sum of the index j in equation (1), and φ ^j (z, x, { It is also possible to calculate only n _q-1 }) and ψ ^j (z, x, {n _q-1 }) and repeat the subsequent calculations (step S220).

In the above ^{operation, φ j (z, x,} {n q-1}) in the case of values and ^{ψ j (z, x, {} n q-1}) and the value of the near, the formula (1) Im [φ ^j (z, x, {n _q-1 }) ^* · ψ ^j (z, x, {n _q-1 })] is a value corresponding to the phase difference, and by reducing this value, It is possible to obtain a desired output.

The determination of the refractive index distribution can be rephrased as defining a virtual mesh in the wave transmission medium and determining the refractive index of a minute region (pixel) defined by the mesh for each pixel. In principle, such a local refractive index can be an arbitrary (desired) value for each location. The simplest system is a system consisting only of pixels having a low refractive index (n _L ) and pixels having a high refractive index (n _H ), and the overall refractive index distribution due to the spatial distribution of these two types of pixels. Is determined. In this case, the place where the low refractive index pixel exists in the medium is considered as a gap of the high refractive index pixel, and conversely, the place where the high refractive index pixel exists is considered as the gap of the low refractive index pixel. Can do. That is, the wave transmission medium of the present invention can be expressed as a desired place (pixel) in a medium having a uniform refractive index replaced with a pixel having a different refractive index.

  The calculation contents for determining the refractive index distribution described above are summarized as follows. An input port and an output port are provided in a medium capable of transmitting a wave in a holographic manner (dielectric in the case of light). Field distribution 1 (forward propagation light) of propagating light incident from the input port and from the input port A field distribution 2 (reverse propagation light) of phase conjugate light in which an output field expected when an incident optical signal is output from the output port is propagated backward from the output port side is obtained by numerical calculation. In the field distribution 1 and the field distribution 2, a spatial refractive index distribution in the medium is obtained so as to eliminate the phase difference at each point (x, z) of the propagating light and the counter propagating light. If the steepest descent method is adopted as a method for obtaining such a refractive index distribution, the refractive index is calculated by changing the refractive index in the direction obtained by the steepest descent method using the refractive index of each point as a variable. By changing as in (1), the difference between the two fields can be reduced. If such a wave transmission medium is applied to an optical component that emits light incident from an input port to a desired output port, an effective optical path length is reduced due to interference phenomenon caused by multiple scattering of propagation waves generated in the medium. An optical circuit having a sufficiently high optical signal controllability can be configured even with a long and gentle refractive index change (distribution).

  FIG. 4 shows the refractive index distribution of the wave transmission medium applied to the arrayed waveguide. According to the algorithm described above, an optical circuit having the refractive index distribution shown in FIG. 4 is obtained by repeating about 200 times. Here, the black part in the optical circuit design region 1-1 in the figure is a high refractive index part (dielectric multiple scattering part) 1-11 corresponding to the core, and the part other than the black part corresponds to the clad. The low refractive index portion 1-12 is a scattering point having a refractive index lower than that of the waveguide. The refractive index of the clad assumes the refractive index of quartz glass, and the refractive index of the core has a value that is higher by 1.5% than the quartz glass. The size of the optical circuit is 250 μm in length and 50 μm in width. The mesh used for the calculation when obtaining the refractive index distribution is 500 × 100.

FIG. 5 shows a method for creating a waveguide including a wave transmission medium. A lower clad glass soot 202 mainly composed of SiO ₂ is deposited on the silicon substrate 201 by flame deposition. Next, a core glass soot 203 obtained by adding GeO ₂ to SiO ₂ is deposited (FIG. 5A). Then, glass transparency is performed at a high temperature of 1000 ° C. or higher. At this time, the glass is deposited so that the lower clad glass layer 204 is 30 microns thick and the core glass 205 is 7 microns thick (FIG. 5B).

Subsequently, an etching mask 206 having a pattern corresponding to the refractive index distribution shown in FIG. 4 is formed on the core glass 205 by using a photolithography technique (FIG. 5C), and the core glass 205 is formed by reactive ion etching. Is patterned (FIG. 5D). After removing the etching mask 206, the upper cladding glass 207 is formed again by the flame deposition method. The upper cladding glass 207 is made of SiO ₂ by adding a dopant such as B ₂ O ₃ or P ₂ O ₅ . As a result, the glass transition temperature is lowered so that the upper clad glass 207 also enters the narrow gap between the core glass 205 and the core glass 205 (FIG. 5E).

  Note that the minimum dimension of the mesh is assumed to be a sufficiently small value as a shape in which light scattering occurs with respect to the wavelength of the optical signal of 1.55 μm. .2 μm.

  FIG. 6 shows a waveguide type optical multiplexing / demultiplexing circuit according to an embodiment of the present invention. As in the waveguide type optical multiplexing / demultiplexing circuit shown in FIG. 1, the arrayed waveguide 106 is divided into three blocks, and the wave transmission medium shown in FIG. 4 is applied to the arrayed waveguide 106b. FIG. 7 shows an enlarged view of the arrayed waveguide 106b. The arrayed waveguide 106b serves as a wave transmission medium, and can be arranged without bending the path of the arrayed waveguide as compared with the prior art. By applying the wave transmission medium, the effective refractive index of the arrayed waveguide can be changed, so that the optical path length of the optical signal propagating through the arrayed waveguide can be shortened, and the waveguide type optical multiplexing / demultiplexing circuit Downsizing can be realized. Compared with a conventional waveguide-type optical multiplexing / demultiplexing circuit, it is possible to realize a reduction in size of about 80% in terms of the substrate length.

  FIG. 8 shows the transmission characteristics of the waveguide type optical multiplexing / demultiplexing circuit shown in FIG. A case where the present invention is applied to an arrayed waveguide diffraction grating type optical multiplexing / demultiplexing circuit having a channel wavelength interval of 20 nm is shown. Good demultiplexing characteristics can be obtained with little effect on the spectrum shape.

It is a figure which shows the structure of the conventional array waveguide diffraction grating type | mold optical multiplexing / demultiplexing circuit. It is a figure for demonstrating the basic structure of a wave transmission medium. It is a flowchart which shows the calculation procedure for determining the spatial refractive index distribution of a wave transmission medium. It is a figure which shows the refractive index distribution of the wave transmission medium applied to an arrayed waveguide. It is a figure which shows the production method of the waveguide containing a wave transmission medium. It is a figure which shows the waveguide type optical multiplexing / demultiplexing circuit concerning one Embodiment of this invention. It is an enlarged view which shows the wave transmission medium of an arrayed waveguide. It is a figure which shows the transmission characteristic of the waveguide type optical multiplexing / demultiplexing circuit concerning one Embodiment of this invention.

Explanation of symbols

DESCRIPTION OF SYMBOLS 100 Substrate 101 Input waveguide 102 First slab waveguide 103, 106 Array waveguide 104 Second slab waveguide 105 Output waveguide

Claims (3)

A first slab waveguide that branches optical signals received from a plurality of input waveguides and a plurality of waveguides that sequentially increase in length by a predetermined waveguide length difference are branched by the first slab waveguide. A waveguide-type optical multiplexing / demultiplexing circuit including an arrayed waveguide for inputting each optical signal, and a second slab waveguide for guiding each signal light output from the arrayed waveguide to each of the plurality of output waveguides In
A wave transmission medium having a spatial refractive index distribution formed in a core layer of the arrayed waveguide in a part of the arrayed waveguide,
The spatial refractive index distribution indicates that when input light incident on the wave transmission medium from the arrayed waveguide propagates along the light propagation direction, the input field of the input light becomes the output field of the output light. Determined by the refractive index of each core of the pixels defined by the mesh to propagate through each set of one or more input / output fields that are transformed and output to the arrayed waveguide;
The refractive index of the core of each of the pixels is the location of the cross section X when the coordinate z in the light propagation direction of the arrayed waveguide is the coordinate x in the direction perpendicular to the light propagation direction in the core layer ( z, in x), so that the phase difference between the and the forward propagation of the field in the input field phase, before Symbol phase field obtained by back propagation to the phase conjugate of the output field is equal to or less than a predetermined error, the The calculation is based on the refractive index distribution {n _q−1 } obtained by the (q−1) th calculation, and the jth input field ψ ^j (x) and output field φ of the set of input / output fields. ^{When j} (x) is the forward propagation field ψ ^j (z, x, {n _q-1 }) and the back propagation field φ ^j (z, x, {n _q-1 }), The refractive index n _q (z, x) at (z, x) is
n _q (z, x) = n _q-1 (z, x) −αΣ _j Im [φ ^j (z, x, {n _q-1 }) ^* · ψ ^j (z, x, {n _q-1 })]
Where the symbol “*” is a complex conjugate, the symbol “•” is an inner product operation, Im [] is the imaginary component of the field inner product operation result in [], and α is the convergence of the calculation Where Σ _j is the sum for j, and
The field of the input light is subjected to multiple phase changes by the refractive index distribution of the cross section X along the propagation direction of the arrayed waveguide, thereby causing multiple scattering of propagation waves generated in the wave transmission medium. A waveguide type optical multiplexing / demultiplexing circuit that propagates while changing the shape of a light field due to an interference phenomenon.   2. The waveguide type optical multiplexer / demultiplexer according to claim 1, wherein the minimum dimension of the pixel is 0.2 [mu] m or more.   3. The waveguide type optical multiplexing / demultiplexing circuit according to claim 1, wherein the waveguide is constituted by a silica glass optical waveguide on a silicon substrate.

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