Optical circuit

Abstract

The present invention provides a novel optical waveguide configuration in which the impact of a fabrication error on optical circuit characteristics is suppressed, while reducing the size of of a waveguide with a high refractive index contrast and maintaining characteristics such as a stable refractive index. An optical circuit according to the present disclosure comprises an optical waveguide that achieves characteristic optical confinement. The optical waveguide includes: a slab-shaped first core; and a second core having a rectangular cross-section. By suppressing the amount of change in the refractive index in directions parallel and perpendicular to the substrate surface, the impact of a manufacturing error on the optical circuit characteristics is mitigated. It is also possible to suppress propagation loss since most of the optical power is confined in the slab-shaped first core, which does not require patterning.

Description

Optical circuit

[0001] The present invention relates to an optical circuit including a waveguide.

[0002] As an optical circuit for ultraviolet / visible light, one using Al2O3 as the core and SiO2 as the cladding has attracted attention (Non-Patent Document 1). Al2O3 and SiO2 have high transmittance in the ultraviolet / visible light region and little change in refractive index over time with respect to ultraviolet / visible light. Since the relative refractive index difference Δ between the core of Al2O3 and the cladding of SiO2 is as large as about 18%, in a channel-type optical waveguide, the propagating light is strongly confined within the core. In a high refractive index difference waveguide with Al2O3 as the core as described above, the optical waveguide can be bent with a small diameter, which is very effective for miniaturizing the size of the optical circuit.

[0003] Ward A. P. M. Hendriks, Dawson B. Bonneville, Soheila Mardani, Meindert Dijkstra and Sonia M. Garcia-Blanco, “UV integrated photonics in sputter deposited aluminum oxide”, Optica Open. Preprint, 2024

[0004] In an optical circuit composed of high refractive index difference waveguides, in order to utilize strong light confinement characteristics, the core width and the gap width between waveguides are often set to a submicron size. In such an optical circuit, due to manufacturing errors of the waveguide, the circuit characteristics of an optical circuit having a directional coupler, a multimode interferometer, etc. were likely to deteriorate. It was difficult to fabricate an optical circuit having desired circuit characteristics with a high yield.

[0005] One embodiment of the present invention is an optical circuit including an optical waveguide in which a lower cladding, a slab-shaped first core above the lower cladding, a second core having a rectangular cross-section perpendicular to the length direction above the first core, and an upper cladding covering the upper surface and both side surfaces of the second core are arranged in this order from a substrate.

[0006] Another embodiment of the present invention is an optical circuit comprising an optical waveguide arranged in this order from a substrate, comprising a lower cladding, a slab-like first core on the lower cladding, an upper cladding covering the first core, the upper cladding having a rectangular cross-section perpendicular to the longitudinal direction and a groove reaching the first core, and a second core formed in the groove.

[0007] In optical circuits including high refractive index difference waveguides, the impact of manufacturing errors on circuit characteristics is mitigated, enabling the production of optical circuits with high yield.

[0008] This figure illustrates the configuration of the optical waveguide in the optical circuit of Embodiment 1 of this disclosure. This figure shows a comparison of the optical waveguide structure between the prior art and the optical circuit of this disclosure. This figure compares the coupling rate variation of the Cr port in the prior art and the optical circuit of this disclosure. This figure illustrates the configuration of the optical waveguide in the optical circuit of Embodiment 2 of this disclosure. This figure shows the optical power distribution in the vertical direction of the optical waveguide in the optical circuit of this disclosure. This figure illustrates the configuration of the optical waveguide in an optical circuit modified from Embodiment 2. This figure shows an example configuration in which another function is added to the optical circuit of Embodiment 2 after flattening.

[0009] The optical circuit disclosed herein provides a novel optical waveguide configuration that minimizes the effects of manufacturing errors while maintaining the characteristics of high refractive index difference waveguides, such as miniaturization and refractive index stability. The optical circuit disclosed herein comprises an optical waveguide that achieves characteristic optical confinement. This optical waveguide includes a slab-shaped first core and a second core having a rectangular cross-section. By suppressing the change in refractive index in directions parallel and perpendicular to the substrate surface, the impact of manufacturing errors on the optical circuit characteristics is mitigated. Since most of the optical power is confined in the slab-shaped first core, which does not require patterning, propagation losses can also be suppressed.

[0010] In the following, the characteristic optical waveguide structure of the optical circuit will be described in comparison with conventional optical circuits including high refractive index difference waveguides. [Embodiment 1] Figure 1 is a diagram illustrating the configuration of an optical waveguide in an optical circuit of Embodiment 1 of the present disclosure. The structure of the optical waveguide will be explained using a directional coupler as an example of the optical circuit 100. Figure 1(a) is a top view of the optical circuit 100 as seen from the substrate surface (x-z plane), (b) is a cross-sectional view perpendicular to the substrate surface passing through the b-b line (x-y plane), and (c) is a cross-sectional view perpendicular to the substrate surface passing through the c-c line (x-y plane).

[0011] The directional coupler achieves the function of combining or splitting propagating light by bringing parts of the two optical waveguides 101a and 101b close together, as shown in Figure 1(a). The optical waveguides 101a and 101b in the optical circuit 100 have the characteristic structures shown in Figures 1(b) and (c).

[0012] The optical waveguide 101b in Figure 1(b), which shows a cross-section perpendicular to the length direction, comprises a lower cladding 10 formed on a substrate (not shown), and a slab-shaped first core 11 formed on the lower cladding 10. Furthermore, it comprises a second core 12 formed on the first core 11, with a rectangular cross-section. An upper cladding 13 is further formed over the entire surface of the first core 11 and the second core 12.

[0013] Even in the region where the two optical waveguides 101a and 101b are in close proximity, as shown in Figure 1(c), each optical waveguide has the same cross-sectional structure as shown in Figure 1(b).

[0014] The specific configurations of the optical waveguides 101a and 101b are as follows: The lower cladding 10 may be an undoped SiO2 layer. The first core 11 may be an Al2O3 layer, as described in the conventional high refractive index difference waveguide. The second core 12 may be a Zr-doped SiO2 layer. The upper cladding 13 may be an undoped SiO2 layer.

[0015] In this embodiment, the thickness of the lower cladding 10 was 20 μm, and the thickness of the first core 11 was 0.2 μm. The second core 12 had a rectangular cross-section patterned with a width of 1.8 μm and a thickness of 2 μm. The thickness of the upper cladding 13 was 1 μm. The Zr doping concentration of the second core 12 was adjusted so that the difference in refractive index between the second core 12 and the upper cladding 13 was 5%.

[0016] The distance between the second cores of the two optical waveguides 101a and 101b shown in Figure 1(c), i.e., the gap width, was set to 1.0 μm. In addition, the length in the z-axis direction (coupled length) of the region where the two optical waveguides 101a and 101b of the directional coupler shown in Figure 1(a) are in close proximity was set to 1 mm.

[0017] The method for fabricating the optical circuit including the optical waveguide shown in Figure 1 is as follows. First, the lower cladding 10, the first core 11, and the second core 12 are deposited on a Si substrate in that order. Next, the second core 12 is patterned by photolithography or electron beam lithography and dry etching, similar to the fabrication process for silica-based optical waveguides. After that, the optical waveguide shown in Figure 1 can be formed by depositing the upper cladding 13.

[0018] Therefore, the optical circuit 100 of the present disclosure can be implemented as an optical waveguide 101 comprising a lower cladding 10, a slab-shaped first core 11 on the lower cladding, a second core 12 on the first core having a rectangular cross-section perpendicular to the longitudinal direction, and an upper cladding 13 covering the top surface and both sides of the second core, arranged in this order from the substrate.

[0019] Figure 2 is a diagram comparing the structure of high refractive index difference waveguides in the prior art and the optical circuit of the present disclosure. Figure 2(a) shows an optical circuit 200 in which a directional coupler is constructed using conventional optical waveguides 201a and 201b, which use Al2O3 as the core and SiO2 as the cladding. High refractive index difference waveguides 201a and 201b are called channel-type optical waveguides. Referring to the A-A cross-sectional view in Figure 2(a), the channel-type optical waveguide has cores 11a and 11b inside the cladding 10, with a rectangular cross-section perpendicular to the length direction of the optical waveguide. Figure 2(b) is identical to the optical circuit 100 of the present disclosure, which has optical waveguides 101a and 10b having the characteristic structure shown in Figure 1.

[0020] Figure 3 shows a comparison of the coupling ratio variation (variation) of the crossport between the conventional technology and the optical circuit of this disclosure. Referring again to Figure 2(b), in a directional coupler, light of different wavelengths input to two input ports (In1, In2) can be combined and output from a single crossport (Cr:Out2). For example, if light with a wavelength of 405 nm is input to In1 and light with a wavelength of 600 nm is input to In2, the two lights are combined and output from the Cr port.

[0021] In the graph of Figure 3, the horizontal axis shows the deviation (μm) of the core width from the design value in the two directional couplers of Figure 2, and the vertical axis shows the coupling rate of light with a wavelength of 405 nm to the Cr port. The core width of the conventional optical circuit 200 is the width of the rectangular cores 11a and 11b, while the core width of the optical circuit 100 of this disclosure is the width of the rectangular second core 12. The detailed structure of the optical waveguides 101a and 101b of the optical circuit 100 of this disclosure is the same as that described in Figure 1(b).

[0022] The channel-type optical waveguides 201a and 201b in Figure 2(a) have Al2O3 cores 11a and 11b and an SiO2 cladding 10. The core width is 0.9 μm and the core thickness is 0.2 μm. The gap width between the two cores 11a and 11b is 0.3 μm and the coupling length is 2 mm. In both configurations of Figure 2(a) and (b), the set core width corresponds to the maximum core width at a wavelength of 405 nm in which the generation of higher-order waveguide modes than the fundamental mode can be substantially ignored.

[0023] As shown in Figure 3, compared to the case of a channel-type optical waveguide, the change in the Cr port coupling ratio in the directional coupler of the optical circuit 100 with respect to core width variation is significantly suppressed. The horizontal axis of the graph in Figure 3 shows the core width variation in absolute value. Compared to the channel type with a core width of 0.9 μm, the optical circuit 100 with a larger core width of 1.8 μm shows a smaller variation in the coupling ratio with respect to core width variation (variation). However, even considering this, the optical circuit 100 of this disclosure has a very small influence on the characteristic value (coupling ratio) from the core width variation of the optical waveguide, i.e., the manufacturing error of the core width. The optical waveguide of the optical circuit 100 of this disclosure has a high tolerance to manufacturing variations, and the variation in optical circuit characteristics is suppressed.

[0024] The difference between this and the conventional high refractive index difference waveguide is that, compared to channel-type optical waveguides, the optical confinement of the optical circuit 100 of this disclosure is weaker in both the width direction (x-axis) and the height direction (x-axis) of the optical waveguide. As mentioned earlier, in the cores 11a and 11b of the channel-type optical waveguides 201a and 201b, the relative refractive index difference Δ between Al2O3 and SiO2 reaches 18%, resulting in strong optical confinement at the boundary between the core and the cladding.

[0025] In contrast, in the optical circuit 100 of this disclosure, the propagating light is confined within the range of the second core by a slab-shaped first core 11 in the width direction (x direction). This optical confinement is thought to be caused by the difference in the relative refractive index Δ between the rectangular second core 12, which is in contact with the first core 11, and the upper cladding 13 on both sides thereof. The refractive index n2 of the Zr-doped SiO2 layer, which is the second core, is in a relationship n2 > n3 with respect to the refractive index n3 of the SiO2 in the upper cladding 13. Therefore, the difference in the relative refractive index Δ between Zr-doped SiO2 and SiO2 remains at about 5%. Compared to the optical waveguides 201a and 201b of the conventional channel-type optical waveguide, which have a relative refractive index difference Δ of 18%, the degree of optical confinement in the optical waveguide of the optical circuit of this disclosure is weaker.

[0026] By weakening optical confinement, the core width satisfying the single-mode condition can be increased, mitigating the impact of manufacturing errors in the core width of the optical circuit on the optical circuit characteristics. At the same time, the gap width between optical waveguides can also be increased, thus easing the requirements imposed on exposure equipment and dry etching equipment. It should be noted here that increasing the core width and gap width in the optical circuit 100 has little effect on miniaturization of the optical circuit. For example, in a directional coupler, the overall size of the optical circuit is determined by the interface portion of the optical input and optical output, and the coupling length, and increasing the core width and gap width at the μm level has little effect. In conventional channel-type high refractive index difference waveguides, the core width could not be made larger than a predetermined value. This is because increasing the core width increases the number of light modes propagating in the waveguide, which can lead to increased circuit losses. In the optical waveguide of the optical circuit 100 of this disclosure shown in Figure 1, most of the optical power is confined within the slab-shaped first core 11, and relatively loose optical confinement occurs due to the smaller specific refractive index difference Δ between the rectangular-shaped second core 12 and the upper cladding 13 on both sides. This relaxation of the degree of optical confinement allows for an increase in core width and gap width, leading to the suppression of variations in optical characteristic values ​​with respect to waveguide width fluctuations, as shown in Figure 3. As will be described later, approximately 85% of the total optical power of the propagating light is confined in the first core portion directly below the second core. Therefore, the characteristic of low change in refractive index over time for ultraviolet / visible light in conventional high refractive index difference waveguides using Al2O3 as the core material is maintained in the optical circuit 100 as well.

[0027] In addition, as a secondary effect, the optical waveguide structure in the optical circuit 100 of this disclosure is expected to have the effect of reducing propagation loss compared to channel-type optical waveguides. The channel-type optical waveguides 201a and 201b with an Al2O3 core shown in Figure 2(a) had the problem of increased propagation loss due to core sidewall roughness generated during the dry etching process. On the other hand, in the optical waveguide structure of the optical circuit 100 of this disclosure, the Al2O3 layer of the first core, through which most of the light propagates, is a slab waveguide and does not require dry etching. Therefore, the loss caused by dry etching, which was a problem in channel-type optical waveguides, does not occur. Furthermore, by forming the upper cladding 13 in the optical waveguide of the optical circuit 100 of this disclosure, the formation of voids in the gap between the two optical waveguides can also be expected to be suppressed.

[0028] In this embodiment, the optical circuit, using a directional coupler as an example, was described to explain the effect of improving the variation in the coupling ratio of Cr ports due to manufacturing errors. The effect of manufacturing errors in the core width and gap of an optical waveguide on the optical circuit characteristics also occurs in other functional circuits that use light propagation and interference. Even for such optical circuits, adopting the configurations of optical waveguides 101a and 101b shown in Figure 1 can provide the same effect of suppressing optical circuit characteristic variations as shown in Figure 3. For example, it is applicable to optical circuits including multimode interferometers (MMI), Mach-Zehnder interferometers (MZI), and array waveguide gratings (AWG).

[0029] The materials for the first core 11 and the second core 12 are not limited to Al2O3 and Zr-doped SiO2 of Embodiment 1. For example, the first core and the second core may be materials with a refractive index higher than or equal to that of undoped SiO2, such as LiNbO3. The refractive index n2 of the second core material must be lower than the refractive index n1 of the first core material (n1 > n2). When undoped SiO2 is used for the second core, the upper cladding may be made of a material with a refractive index lower than that of undoped SiO2, such as fluorine-doped SiO2 or MgF2. In addition to undoped SiO2, the cladding material may also be fluoride materials such as fluorine-doped SiO2, MgF2, BaF2, or LaF3.

[0030] In Embodiment 1 described above, the lower cladding 10 was described as being formed separately on the Si substrate. However, if undoped SiO2 is used as the substrate material, the substrate itself can also be used as the lower cladding 10. [Embodiment 2] In the optical waveguide of the optical circuit of Embodiment 1, most of the total optical power of the propagating light is distributed within the first core. Optical confinement parallel to the substrate surface (in the x-axis direction) is controlled by a second core that is patterned in a rectangular shape. This optical confinement is caused by the refractive index difference between the second core and the upper cladding on both sides thereof. Similar optical confinement can also be achieved by other optical waveguide configurations that are suitable for further integration.

[0031] Figure 4 is a diagram illustrating the configuration of an optical waveguide in the optical circuit of Embodiment 2 of the present disclosure. Figure 4(a) shows the structure of the optical waveguide 101 of Embodiment 1 for comparison and is identical to Figure 1(b), so a detailed explanation is omitted. Figure 4(a) also schematically shows the optical confinement range 18. Figure 4(b) shows the structure of a cross-section (x-y plane) perpendicular to the longitudinal direction of the optical waveguide of Embodiment 2.

[0032] The optical waveguide 102 of Embodiment 2 comprises a lower cladding 10 formed on a substrate (not shown), and a slab-shaped first core 11 formed on the lower cladding 10. Furthermore, an upper cladding 13 is provided on the first core 11, except for the portion that will become the optical waveguide 102. As will be described later, the portion that will become the optical waveguide 102 is formed by removing the upper cladding 13 by patterning, creating a groove that exposes the upper surface of the first core 11. A second core 14 is formed on the entire surface of the upper cladding 13, where a groove with a rectangular cross-section is formed.

[0033] The groove from which the material of the upper cladding 13 has been removed is filled with a second core 14. The configuration that contributes to light confinement in the first core 11 is the same in Embodiment 1 shown in Figure 4(a) and Embodiment 2 shown in Figure 4(b), as will be described later.

[0034] The specific configuration of the optical waveguide 102 is as follows: The lower cladding 10 may be an undoped SiO2 layer. The first core 11 may be an Al2O3 layer, similar to Embodiment 1. The upper cladding 13 may also be an undoped SiO2 layer, similar to Embodiment 1. Similar to Embodiment 1, the second core 14 may be a Zr-doped SiO2 layer, and the upper cladding 13 may be an undoped SiO2 layer.

[0035] In this embodiment, the thickness of the lower cladding 10 was 20 μm, the thickness of the first core 11 was 0.2 μm, and the thickness of the second core 14 was 2 μm. As described above, a groove reaching the upper surface of the first core 11 was formed in the upper cladding 13, with a groove width of 1.8 μm and a depth of 2 μm. The Zr doping concentration of the second core 14 was adjusted so that the difference in relative refractive index between the Zr-doped SiO2 second core 14 and the undoped SiO2 upper cladding 13 was 5%.

[0036] The method for fabricating the optical waveguide of the optical circuit of Embodiment 2 shown in Figure 4(b) is as follows. First, the lower cladding 10, the first core 11, and the upper cladding 13 are deposited on a Si substrate in that order. Next, similar to the fabrication process for silica-based optical waveguides, the upper cladding 13 is removed by patterning using photolithography or electron beam lithography and by dry etching to form grooves. After that, the optical waveguide 102 shown in Figure 4(b) can be fabricated by depositing the second core 14 over the entire surface.

[0037] Therefore, the optical circuit of the present disclosure can also be implemented as an optical waveguide comprising a lower cladding 10, a slab-shaped first core 11 on the lower cladding, an upper cladding 13 covering the first core, having a rectangular cross-section perpendicular to the longitudinal direction and a groove reaching the first core, and a second core 14 formed in the groove, arranged in this order from the substrate.

[0038] Figure 5 shows the vertical optical power distribution in the optical waveguide of the optical circuit of the present disclosure. Figure 5(a) shows the optical power distribution in the direction perpendicular to the substrate surface (y-axis) at the center position x=0 of the optical waveguides of Embodiments 1 and 2. Figure 5(b) shows the correspondence between the y-axis position on the horizontal axis of the graph in (a) and the configuration of the optical waveguide of Embodiment 1.

[0039] It can be seen that the two optical power distributions, the solid line in Embodiment 1 and the dotted line in Embodiment 2 in Figure 5(a), are identical. This is because the surrounding refractive index distribution that determines the region of the "optical confinement range 18" shown in Figure 5(b), where the propagating light is distributed, is the same in both embodiments.

[0040] Referring again to Figure 4, in the direction perpendicular to the substrate surface (y-axis direction), the optical confinement range 18 is determined by the refractive index difference between the lower cladding 10 and the first core 11, and the refractive index difference between the first core 11 and the second core 12. In the direction parallel to the substrate surface (x-axis direction), in Embodiment 1, the optical confinement range 18 is determined by the refractive index difference between the second core 12 and the upper cladding 13 on both sides. In Embodiment 2, the optical confinement range 18 is determined by the refractive index difference between the second core 14 in the groove and the upper cladding 13 on both sides.

[0041] Since the shapes that demarcate the regions of the lower cladding, the first core, the second core, and the upper cladding are exactly the same in both embodiments, it can be understood that the two optical power distributions in Figure 5(a) are identical.

[0042] As is clear from the optical power distribution in Figure 5(a), at the outermost surface position in the y-axis direction (y = 2.2), the second cores 12 and 14 have no effect whatsoever on the effective refractive index of the propagating light in the confinement range 18. In the configuration of Embodiment 2 shown in Figure 4(b), the portion of the upper cladding 13 near the outermost surface is also unnecessary for optical confinement. The minimum height to be left as the second core can be determined by determining the thickness of the second core such that 95% of the total optical power is contained within the first and second cores in the direction perpendicular to the substrate surface of the optical waveguide, and this thickness should be set as the minimum height. By making the thickness of the second core greater than this minimum height, it is possible to avoid the optical circuit characteristics being affected by dust and other particles adhering to the outermost surface of the optical circuit.

[0043] Figure 6 illustrates the configuration of an optical waveguide in an optical circuit modified from Embodiment 2. Figure 6(a) is identical to the configuration of Embodiment 2 shown in Figure 4(a). In the portion of the optical circuit above the dotted line, which is a certain distance from the first core, the effective refractive index of the propagating light is not affected, and the optical power of the propagating light is negligibly small. By removing this portion by polishing such as CMP, it is possible to flatten the outermost surface of the optical circuit as shown in Figure 6(b). In the configuration of the optical waveguide in Embodiment 2, the thickness (y-axis direction) of the second core 14 in the groove is sufficient if it is 1.5 μm or more from the first core 11. For further flattening, another material may be deposited on top of the second core after removal. Alternatively, to ensure wafer-level flatness, another material may be deposited before CMP, and then CMP may be performed. By going through the flattening process, other functions can be integrated on the upper side of the optical waveguide.

[0044] FIG. 7 is a diagram showing a configuration example in which another function is added to the optical circuit of Embodiment 2 after planarization. FIG. 7(a) shows a top view when a Mach-Zehnder interferometer (MZI) is configured in the optical circuit 110. The two optical waveguides 103a and 103b are arm waveguides, and a heater 16 is formed on one of the arm waveguides. By heating the heater 16, the interference state of the MZI is changed, and an optical signal processing function such as an optical switch is realized. FIG. 7(b) is a cross-sectional view taken perpendicular to the substrate surface along line A-A. The cross-sectional structures of the two optical waveguides 103a and 103b are the same as those in FIG. 6(b), and since the outermost surface is planarized, the heater 16 can be stably fabricated.

[0045] In the optical waveguides of the optical circuits of Embodiment 2 in FIGS. 4, FIG. 6(b), and FIG. 7, the second core 14 is formed by first removing a part of the upper cladding 13 to form a groove and depositing to fill the groove. Therefore, there is no need to dry-etch the second core as in the configuration of Embodiment 1. It becomes easy to use materials that are difficult to dry-etch, such as fluoride, LiNbO3, and SiO2 with a high concentration of ZrO2, as the material of the second core.

[0046] As described in detail above, the optical circuit of the present disclosure provides a novel configuration that suppresses the influence of manufacturing errors on the optical circuit characteristics by a characteristic optical waveguide with weaker optical confinement compared to a channel-type optical waveguide.

[0047] The present invention can be used for optical communication and optical signal processing.

Claims

1. An optical waveguide comprising an optical waveguide in which a lower cladding, a slab-shaped first core on the lower cladding, a second core on the first core having a rectangular cross-section perpendicular to the longitudinal direction, and an upper cladding covering the top surface and both sides of the second core are arranged in this order from a substrate.

2. An optical circuit comprising an optical waveguide arranged in this order from a substrate: a lower cladding, a slab-shaped first core on the lower cladding, an upper cladding covering the first core, having a rectangular cross-section perpendicular to the longitudinal direction and a groove reaching the first core, and a second core formed in the groove.

3. The optical circuit according to claim 1 or 2, wherein the lower cladding is the substrate or formed on the substrate.

4. The optical circuit according to claim 1 or 2, wherein n1 is the refractive index of the first core and n2 is the refractive index of the second core, and n1 > n2.

5. The optical circuit according to claim 1 or 2, wherein the thickness of the second core is set such that 95% of the total optical power is contained within the first core and the second core in the direction perpendicular to the substrate of the optical waveguide.

6. The optical circuit according to claim 1 or 2, wherein the first core is composed of Al2O3, LiNbO3, BaF2, or LaF3.

7. The optical circuit according to claim 1 or 2, wherein the second core is composed of any of Zr-doped SiO2, undoped SiO2, Hf-doped SiO2, BaF2, or LaF3.

8. The optical circuit according to claim 1 or 2, wherein optical confinement in a direction perpendicular to the substrate is determined by the difference in specific refractive index between the lower cladding and the first core, and the difference in specific refractive index between the first core and the second core, and optical confinement in a direction parallel to the substrate is determined by the difference in refractive index between the second core and the upper cladding on both sides.

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