JP2001074955A - Photonic crystal waveguide - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a low-loss small-size waveguide using a three-dimensional photonic crystal. SOLUTION: The photonic crystal has stripe layers each having a plurality of linear bodies periodically arranged parallel to one another at a specified pitch in such a manner that the linear bodies in the n-th layer from one side are perpendicular to the linear bodies in the (n+1)-th layer, and linear bodies in the n-th layer are parallel to the linear bodies in the (n+2)-th layer, and the arrangement of the linear bodies in the n-th layer is shifted by a half of the specified pitch from that of the linear bodies in the (n+2)-th layer. In one layer of the photonic crystal not at either end of the crystal, at least part of at least one linear body is absent to form a linear defect, and the linear defect functions as a waveguide.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to a photonic crystal waveguide in which a waveguide is formed by providing a defect in a three-dimensional photonic crystal.

[0002]

2. Description of the Related Art In recent years, a material called a photonic crystal has attracted attention (eg, Physical Review Letter).
s vol.58.p2059-2062 (1987) (Reference 1), Japanese Jo
urnalof Applied Physics.vol.35, p909-912 (1996)
(Reference 2)). A photonic crystal is a crystal that has a periodic refractive index distribution inside it and forms a photonic band gap to control spontaneous emission light.
It is possible to freely control light in a minute area.

When a defect is linearly introduced into a photonic crystal having a photonic band gap, light is guided along the defect and can be used as a waveguide.

An example of forming a waveguide by introducing a defect into a photonic crystal is described in Physical Review Letters.
l.77, p3787-3790 (1996) (Reference 3) and Proceedings of the 46th Joint Lecture on Applied Physics, 29p-E-11 (1999) (Reference 4). Reference 3 analyzes a waveguide formed by introducing a linear defect into a two-dimensional photonic crystal.
This shows that high transmission characteristics can be obtained even with a bending of 90 °. Reference 4 analyzes a waveguide formed by introducing a linear defect into a three-dimensional photonic crystal, and shows that an electric field concentrates on the defect.

[0005]

It is well known that a waveguide is formed by introducing a linear defect into a photonic crystal, but it is still unclear what defect is effective to introduce a waveguide. Not well considered. A two-dimensional photonic crystal is studied as shown in Reference 3, but a three-dimensional crystal, which is expected to have a waveguide with little loss because it is a perfect crystal, is not specifically studied. . Literature 4 analyzes a linear waveguide introduced into a three-dimensional photonic crystal, but does not discuss a specific method for introducing defects. The analysis in Document 4 is only for a linear waveguide,
A waveguide with a bent or branched structure has not been analyzed.

The present invention has been made under such circumstances. An object of the present invention is to provide a low-loss and small-sized waveguide using a three-dimensional photonic crystal.

[0007]

The above object is achieved by the following (1).
This is achieved by the present invention of (4). (1) It has a stripe layer in which a plurality of linear bodies composed of a substance having a higher refractive index than air are periodically arranged in parallel with each other at a constant pitch, and a plurality of such stripe layers are laminated. The linear body in the nth stripe layer from the side is orthogonal to the linear body in the (n + 1) th stripe layer, and the linear body in the nth stripe layer and the linear body in the (n + 2) th stripe layer from one side. A photonic crystal in which the body is parallel and the arrangement of the linear bodies in the n-th stripe layer is shifted from the arrangement of the linear bodies in the (n + 2) -th stripe layer by half of the predetermined pitch. At least a portion of at least one linear body is missing in at least one stripe layer that is not located at either one end of the photonic crystal and the other end thereof; A photonic crystal waveguide in which a linear defect is formed, and the linear defect functions as a waveguide. (2) The linear defect exists in each of a pair of adjacent stripe layers, and the linear defect of one stripe layer and the linear defect of the other stripe layer overlap each other to form a 90 ° bending waveguide or a branch waveguide. The photonic crystal waveguide according to (1) above, wherein (3) The photonic crystal waveguide according to the above (2), which has a spiral waveguide composed of a plurality of the 90 ° bent waveguides. (4) Five or more stripe layers exist between the stripe layer having the linear defect and one end of the photonic crystal and the other end thereof.
The photonic crystal waveguide according to any one of (1) to (3).

[0008]

DESCRIPTION OF THE PREFERRED EMBODIMENTS In general, a photonic crystal is an optical material having a periodic refractive index distribution therein. In a photonic crystal, a photonic band gap is formed with respect to the energy of photons on the same principle as a band gap is formed by a periodic potential distribution composed of atomic nuclei in a solid crystal.
As a result, light having photon energy corresponding to the photonic band gap does not propagate through the photonic crystal. Photonic crystals having such a function are expected to be applied to ultra-small optical circuits and optical devices.

The one having a one-dimensional periodic refractive index distribution has long been known as a multilayer thin film, and is not called a photonic crystal. The characteristics of the nick crystal begin to appear, and by taking a structure having a three-dimensional refractive index distribution, a perfect crystal can be formed. This perfect crystal is
It is a structure capable of confining light in all directions, and has a diamond structure or an asymmetric face-centered cubic structure with a three-dimensional refractive index distribution. In the present invention, an optical waveguide is formed by using a three-dimensional photonic crystal and providing defects in the crystal. For an overview of the three-dimensional photonic crystal used in the optical waveguide of the present invention, see, for example, IEICE Journal Vol.82, No.3, pp.232-241 (1999) (Reference 5), Optics, Vol. No. 1, pp. 6-11 (1998) (Reference 6).

FIG. 1A shows a configuration example of the photonic crystal waveguide of the present invention. This photonic crystal waveguide is obtained by forming a waveguide in a three-dimensional photonic crystal. The three-dimensional photonic crystal is formed by three-dimensionally stacking a stripe layer composed of an air / semiconductor diffraction grating, that is, a stripe layer in which a plurality of linear bodies arranged in parallel to each other are arranged at a constant pitch. It forms an asymmetric face-centered cubic structure. The asymmetric face-centered cubic structure means a face-centered cubic structure having asymmetric lattice points. The lattice points in the three-dimensional photonic crystal are intersections of linear bodies.

In this three-dimensional photonic crystal, the n-th stripe layer 2 from one side in the stripe layer stacking direction
a linear body in the _n, and a linear body in the n + 1 th stripe layer 2 _{n + 1} are orthogonal. Further, a linear body in the striatum and n + 2 th stripe layer 2 _{n + 2} in the n-th stripe layer 2 _n are parallel, and the sequence of the linear body in the n-th stripe layer 2 _n is, It is shifted from the arrangement of the linear bodies in the (n + 2) th stripe layer 2 _{n + 2} by half of the predetermined pitch.

[0012] Thus, by the n-th arrangement period of the linear body with respect to the arrangement period of the linear body in the stripe layer 2 _n in the n + 2 th stripe layer 2 _{n + 2} half phase shifted, asymmetric surface A centered cubic structure is formed. That is, the {001} plane of the face-centered cubic structure is formed by laminating two orthogonal stripe structures, and further,
By shifting and stacking the {001} planes as described above, asymmetry is introduced into the face-centered cubic structure. In this photonic crystal, since the same structure is repeated with four stripe layers as one unit, the number of stacked stripe layers is usually an integral multiple of four. In the illustrated example, only one unit of the photonic crystal is displayed, and the display of the other stripe layers is omitted.

The asymmetric face-centered cubic structure is used in order to form a photonic band gap in all light propagation directions, that is, to obtain a perfect crystal. Not all periodic structures having a three-dimensional refractive index distribution have a photonic band gap. In order to form a photonic band gap, a diamond structure or an equivalent asymmetric face-centered cubic structure is generally used. It is considered necessary.

In order to provide a waveguide in such a three-dimensional photonic crystal, it is necessary to form a linear defect in the crystal. In the present invention, as shown in FIG. 1A, a linear defect is formed by providing a defective portion 3 in one of the linear bodies arranged periodically. Defective part 3 in the figure
Is displayed as a white linear body, but this white part is actually air as well as its surroundings. In the drawings, the traveling direction of the electromagnetic wave is indicated by an arrow.

By the way, it is known that a waveguide can be formed by providing a linear defect in a three-dimensional photonic crystal using a laminated structure of an air / semiconductor diffraction grating as described in the above-mentioned document 4, for example. ing. However, conventionally, no specific study has been made on how to provide a linear defect. As shown in FIG. 1, the method of forming a linear defect by deleting one linear body has one electric field mode existing in the waveguide.
That is, there is an advantage that a single mode can be obtained.

FIG. 1B shows another configuration example of the photonic crystal waveguide. This waveguide is provided with a plurality of linear members having a defective portion 3 having a length substantially equal to the arrangement pitch of the linear members, and the plurality of defective portions 3 are arranged in a row in a direction orthogonal to the linear member extending direction. Are arranged. Also in this configuration, the same operation and effect as the configuration in FIG. The length of the defective portion 3 shown in FIG. 1 (B) is most preferably one time the arrangement pitch of the linear bodies, but is usually 0.6 to 1.4 times the arrangement pitch, preferably 0.1 mm. 8 ~
If it is 1.2 times, there is no problem. If the length is too short, the guided mode may not exist, and if it is too long, the multimode may exist.

According to the present invention, 90%
° Bending waveguides or branching waveguides can be formed. FIG. 2 shows a configuration example of a 90 ° bending waveguide. In this configuration example, in one stripe layer 2, FIG.
One linear body having a long defect 31 similar to the configuration of FIG. 1 is provided to form the first waveguide, and a plurality of linear bodies having a short defect 32 similar to the configuration of FIG. The plurality of defective portions 32 are arranged in a line in a direction perpendicular to the extending direction of the linear body to form a second waveguide. In this configuration example, the first waveguide and the second waveguide are 90
° constitutes a bent waveguide. In the illustrated example, if one or both of the two orthogonal waveguides are extended, a T-shaped branch waveguide (two-branch waveguide) or an X-shaped branch waveguide (three-branch waveguide) is formed. be able to.

In FIG. 2, since two orthogonal waveguides are formed in one stripe layer, the structures of the two waveguides are different. Therefore, the waveguide mode differs before and after bending or before and after branching. As a result, electromagnetic waves are not smoothly coupled, and the loss due to reflection at the time of bending or branching increases. In order to suppress such loss, the 90 ° bending waveguide has the configuration shown in FIG. 3, the T-shaped branch waveguide has the configuration shown in FIG. 4 or FIG. 5, and the X-shaped branch waveguide has the configuration shown in FIG. It is preferable that FIGS. 3 to 6 are plan views showing only two adjacent stripe layers in order to make it easy to see the defective portion provided in the linear body. (A) in each of these figures is:
As in the configuration example of FIG. 1A, a linear defect is formed by providing a defective portion in at least a part of one linear body. FIGS. Similar to the configuration example of B), a linear defect is formed by providing a plurality of linear bodies having a defective portion having substantially the same length as the arrangement pitch.

In the configuration examples shown in FIGS.
A pair of stripe layers 2 in contact_n, 2_{n +1}The smell of each
And at least a part of the linear body_n, 3_{n + 1}Provided
To form a linear defect, and the one stripe layer 2
_nMissing part 3_nAnd the other stripe layer 2_{n + 1}Missing part 3_{n + 1}
And have repeated. Thus, a pair of adjacent stripes
In the loop layer, the intersection of two linear bodies
By removing the difference part, cross-shaped defects in the uneven state
If formed, linear defects in one of the stripe layers can be advanced.
At least some of the electromagnetic waves that have changed
Then, the process proceeds to the linear defect of the other stripe layer. Accordingly
Therefore, in the waveguides shown in FIGS.
The loss associated with branching can be significantly reduced.

The T-shaped branch waveguide shown in FIG. 4 and the T-shaped branch waveguide shown in FIG. 5 have exactly the same structure, and differ only in the branching direction of the electromagnetic wave. That is, in FIG. 4, the electromagnetic wave that has traveled along the first waveguide formed on the stripe layer 2 _{n + 1} or 2 _n travels on the second waveguide formed on the stripe layer 2 _n or 2 _{n + 1.} 180 ° in the second waveguide
It is bifurcated in different directions. Meanwhile, the electromagnetic wave has progressed a first waveguide formed in a stripe layer 2 _{n + 1} or 2 _n in FIG. 5, a portion of proceeds as it first waveguide, the remaining stripe layer 2 _n or By going to the second waveguide formed at 2 _{n + 1} , the light is branched into two. 4 and 5, the branching ratio is almost 1: 1.
However, the configuration of FIG. 4 is preferable because the efficiency is higher.

In the case of forming the 90 ° bending waveguide shown in FIG. 3A, the length of each of the missing portions protruding from the intersection of the two missing portions is 0.5 times the arrangement pitch of the linear bodies. It is most preferable, but usually 0.3
There is no problem if it is 0.7 to 0.7 times, preferably 0.4 to 0.6 times. If the length is too short or too long, the transmittance will be attenuated in a specific wavelength range.

Further, when the T-shaped branch waveguide shown in FIGS. 4A and 5A is formed, the length of the protruding portion corresponding to the T-shaped vertical bar from the intersection of the two portions is as follows. , FIG.
It is preferable that the length is the same as the protruding length in the 90 ° bending waveguide shown in FIG. The reason for the limitation is 90
The same applies to the case of a bent waveguide.

FIG. 7 shows a configuration example of a spiral waveguide. This spiral waveguide has a 90 ° angle of the structure shown in FIG. 3A by providing a defective portion 3 in each of four continuous stripe layers and overlapping the defective portions 3 between adjacent stripe layers. It is formed by connecting three bending waveguides. In this structure, each time the electromagnetic wave passes through the 90 ° bending waveguide once, it moves one layer in the stripe layer. Therefore, in this spiral waveguide, three-dimensional traveling of an electromagnetic wave, which is impossible with a two-dimensional photonic crystal, is possible. Note that the number of connected 90 ° bending waveguides may be determined as needed. Such a spiral waveguide is shown in FIG.
This can also be realized by connecting a plurality of 90 ° bending waveguides shown in FIG.

In the present invention, in order to take advantage of the features of the three-dimensional photonic crystal, it is necessary that the stripe layer for introducing a linear defect is not located at either one end or the other end of the photonic crystal. That is, at least one, and preferably five or more, stripe layers are present between the stripe layer having the linear defect and each end of the photonic crystal. If the number of the stripe layers existing on both sides of the stripe layer in which the linear defect is introduced is too small, the loss of the waveguide increases. In FIGS. 1 to 7, a stripe layer having a linear defect is shown as the uppermost layer in order to clearly show the linear defect. However, a stripe layer is further laminated thereon.

Various parameters in the photonic crystal, for example, the refractive index, dimensions and arrangement pitch of the linear body are not particularly limited. Since the photonic band gap changes depending on these parameters, these parameters may be appropriately determined according to the wavelength to which the waveguide is applied.

In the above, the constituent material of the linear body has been described as a semiconductor. However, the constituent material of the linear body may be any substance having a refractive index higher than that of air, such as a semiconductor, a dielectric, or a conductor. Any of them may be used. As a semiconductor material, for example, Ga
III-V compound semiconductors such as As and GaP, and I such as CdTe
Any of an I-VI group compound semiconductor, a group IV semiconductor such as Si, Ge and the like may be used. However, when the photonic crystal waveguide of the present invention is used by being incorporated in a part of an optical circuit or an optical device, or when an optical circuit or an optical device in addition to the waveguide is integrally formed in the photonic crystal. The material for the linear body is appropriately selected in consideration of the components other than the waveguide. Also, by appropriately selecting the constituent materials,
A photonic crystal can be easily manufactured.
For example, when a manufacturing method described later is used, it is preferable to use a semiconductor material containing Ga and / or As because there are many applicable etching processes.

The refractive index of the linear material is preferably 2
The number is more preferably 3 or more. If the refractive index of the linear material is too low, it is difficult to form a photonic band gap.

Next, a preferred method which can be used for manufacturing the waveguide of the present invention will be described with reference to FIG. In addition,
For the three-dimensional photonic crystal used in the present invention, a part of the manufacturing method is described in the above-mentioned documents 5 and 6.

In the step shown in FIG. 8A, an AlGaAs layer 11 and a Ga film are formed on a GaAs substrate 10 by, for example, MBE at a substrate temperature of, eg, 600 ° C.
The As layer 12 is grown in this order. The AlGaAs layer 11 later becomes an etching stop layer, and the GaAs layer 12 later becomes a stripe layer.

In the step shown in FIG. 8B, the GaAs layer 12 is processed into a stripe shape using photolithography and RIE (Reactive Ion Etching) to obtain a stripe layer 2. At this time, if necessary, patterning is performed so that at least a part of at least one linear body is lost, and a linear defect is introduced. As the reaction gas, for example, an H ₂ + CH ₄ mixed gas can be used.

In the step shown in FIG. 8C, two substrates 10 each having the exact same stripe layer 2 manufactured through the above steps are overlapped so that the stripe layers 2 are in contact with each other. Join them together. Prior to this joining, the joining surface is preliminarily BHF (buffered H
F) It is preferable to perform pretreatment by dipping in a solution. At the time of joining, for example, 650 in an H ₂ atmosphere, for example.
It is preferable to fuse the stripe layers to each other by heating to about ° C.

Next, for example, by removing one of the substrates 10 according to the following procedure, as shown in FIG.
It is assumed that two stripe layers 2 are stacked on the substrate 10. First, in the state shown in FIG.
Is polished to make it thinner. Subsequently, for example, NH ₄ OH
The polished substrate 10 is selectively dissolved by an etching solution in which the solution and the H ₂ O ₂ solution are mixed. At this time, the AlG existing between the substrate 10 and the stripe layer 2
The aAs layer 11 functions as an etching stop layer. Next, the AlGaAs layer 11 is removed with an HF solution.

The substrates 10 thus obtained in the state shown in FIG. 8D are overlapped so that the uppermost stripe layers 2 are in contact with each other, and the stripe layers are joined. However, at this time, in order to introduce the asymmetry described above,
It is necessary to shift the arrangement of the linear bodies parallel to each other on one substrate side and the other substrate side. At this time, a method using a laser beam diffraction pattern is preferable for alignment of the stripe layers to be joined. Specifically, when laser beams are incident on parallel diffraction gratings,
This is based on the fact that the intensity of the ± 1st-order diffraction spot changes according to the positional relationship between the two diffraction gratings, and the intensity of the spot becomes weak again when the positions of the two diffraction gratings are shifted by exactly half a cycle.

After the stripe layers are joined together, one of the substrates 10 is removed by the above-described procedure, and as shown in FIG. 8E, a state in which four stripe layers 2 are laminated on the substrate 10 is obtained. .

By repeating the above steps, it is possible to obtain a photonic crystal waveguide in which an arbitrary number of units are stacked with four stripe layers as one unit.

Next, the characteristics of a specific example of the photonic crystal waveguide of the present invention will be described.

The linear body is made of GaAs, and the lattice constant of the photonic crystal (the arrangement pitch of the linear bodies) is a.
The width (dimension in the arrangement direction) of the linear body was 0.25a, and the height was 0.3a. When no linear defect is formed in the photonic crystal, the normalized frequency c / a (c is the speed of light)
Formed a band gap between 0.380 and 0.444.

On the other hand, as shown in FIG. 1A, by removing one linear body, a time domain difference (FD) is obtained in the case where a linear defect extending in the X-axis direction of the crystal is provided.
When analyzed by the TD) method, an electromagnetic field pattern was obtained in the band gap. The waveguide mode at this time was only the mode in which the electric field was polarized parallel to the Z axis (the axis perpendicular to the stripe layer), and the mode of the electric field polarized parallel to the X axis was not guided. From this result, it was confirmed that the waveguide was formed by providing the linear defect.

FIG. 9 shows the dispersion relationship of k-ω in k-space in the photonic crystal waveguide. From the figure, it can be seen that the mode exists over a wide frequency range.
It can be seen that a single mode is formed and the linearity is relatively good.

Even when the structure shown in FIG. 1B is used, almost the same results as in the case of the structure shown in FIG. 1A are obtained.

Analysis of the 90 ° bending waveguide shown in FIG. 3A shows that electromagnetic waves smoothly couple in the same mode between upper and lower linear waveguides that intersect at different levels.
° The turn progressed. In this waveguide, the electric field intensities before and after bending by 90 ° were almost the same. That is, it was found that the transmittance of the waveguide was almost 1, and the loss could be almost completely suppressed. Note that even in the case of the structure shown in FIG.
Almost the same results were obtained.

On the other hand, when the 90 ° bending waveguide shown in FIG. 2 was analyzed, the waveguide modes did not match before and after the 90 ° bending. Therefore, electromagnetic waves do not couple smoothly,
The transmittance of the waveguide was about 0.5.

When the T-shaped branch waveguides shown in FIGS. 4 and 5 are analyzed, the electromagnetic wave has a branching ratio of about 1:
Branched into two at a ratio of 1.

When the X-shaped branch waveguide shown in FIG. 6 was analyzed, the electromagnetic wave was branched into three at a branching ratio of approximately 1: 1: 1.

[0045]

According to the present invention, a low-loss waveguide can be realized by using a three-dimensional photonic crystal having a photonic band gap. Further, according to the present invention, a bent structure and a branch structure useful for applying the waveguide to various optical circuits can be introduced into the photonic crystal. In a normal waveguide, the radius of curvature is increased to minimize bending loss. However, in a photonic crystal, light is cut off due to the presence of a photonic band gap, so that it can be bent sharply. Therefore, in the present invention,
The size of the optical circuit can be significantly reduced. Also,
In the present invention, if the waveguide is formed into a branch waveguide, an ultra-small optical branch is realized.

[Brief description of the drawings]

FIGS. 1A and 1B are perspective views showing a configuration example of a photonic crystal waveguide of the present invention.

FIG. 2 is a perspective view showing a configuration example in which a 90 ° bending structure is introduced into the photonic crystal waveguide of the present invention.

FIGS. 3A and 3B are plan views showing a configuration example in which a 90 ° bending structure is introduced into the photonic crystal waveguide of the present invention.

FIGS. 4A and 4B are plan views showing a configuration example in which a T-shaped branch structure is introduced into the photonic crystal waveguide of the present invention.

FIGS. 5A and 5B are plan views showing a configuration example in which a T-shaped branch structure is introduced into the photonic crystal waveguide of the present invention.

FIGS. 6A and 6B are plan views showing a configuration example in which an X-shaped branch structure is introduced into the photonic crystal waveguide of the present invention.

FIG. 7 is a perspective view showing a configuration example in which a spiral structure is introduced into the photonic crystal waveguide of the present invention.

FIGS. 8A to 8E are perspective views illustrating a method for manufacturing a photonic crystal waveguide according to the present invention.

FIG. 9 is a graph showing a k-ω dispersion relationship in a k-space in a photonic crystal.

[Explanation of symbols]

_{2,2 n, 2 n + 1,} 2 n + 2 stripe layer _{3,3 n, 3 n + 1,} 31,32 defect

 ────────────────────────────────────────────────── ─── Continuing on the front page (72) Inventor Chutinan Aron Khan 232-403, Upper Capital Letter Town, Takeya-cho, Kawaramachi, Nakagyo-ku, Kyoto (72) Inventor Daisuke Miyauchi 1-13-1 Nihonbashi, Chuo-ku, Tokyo TDK Corporation (72) Inventor Yoshikazu Narimiya 1-13-1 Nihonbashi, Chuo-ku, Tokyo TDK Corporation F-term (reference) 2H047 KA01 KA12 KA12 LA12 QA02

Claims (4)

[Claims] 1. A plurality of linear bodies composed of a substance having a refractive index higher than that of air have a stripe layer which is periodically arranged at a constant pitch in parallel with each other. The linear body in the n-th stripe layer from one side is orthogonal to the linear body in the (n + 1) -th stripe layer, and the linear body in the n-th stripe layer from one side and the linear body in the (n + 2) -th stripe layer A photonic crystal in which the linear bodies are parallel to each other and the arrangement of the linear bodies in the n-th stripe layer is shifted from the arrangement of the linear bodies in the (n + 2) -th stripe layer by half of the predetermined pitch. And at least a portion of at least one linear body in at least one stripe layer not located at one end of one or the other of the photonic crystal. Deficient linear defects are formed, the photonic crystal waveguide which this linear defect functions as a waveguide. 2. A linear defect exists in each of a pair of adjacent stripe layers, and the linear defect in one stripe layer and the linear defect in the other stripe layer overlap each other and form an angle of 90 °.
2. The photonic crystal waveguide according to claim 1, wherein the photonic crystal waveguide forms a bent waveguide or a branch waveguide. 3. The photonic crystal waveguide according to claim 2, further comprising a helical waveguide including a plurality of said 90 ° bent waveguides. 4. The photonic crystal according to claim 1, wherein five or more stripe layers are present between the stripe layer having the linear defects and one end of the photonic crystal. Photonic crystal waveguide.