JP4954828B2 - Optical waveguide circuit and manufacturing method thereof - Google Patents

Description

  The present invention relates to an optical waveguide circuit including a spot size converter.

  Due to the worldwide spread of Internet use, optical communication systems using optical wavelength division multiplexing (WDM) technology that can transmit large volumes of data such as images and video information at high speed are also being introduced into the metro network. . Along with this, research and development of optical circuits constituting optical communication systems has been spurred, and waveguide type optical circuits that can form optical waveguides on a flat substrate by applying LSI microfabrication technology are integrated. It is expected to be a means for realizing high-performance and complex optical circuits because of its excellent performance and mass productivity. In particular, silica-based optical waveguides have low loss and high reliability, so various modules have been put to practical use.

  In the optical waveguide circuit, a wide variety of optical circuits can be realized by functionally utilizing the light interference phenomenon. Among them, the optical wavelength multiplexer / demultiplexer is important as a key device of the WDM system.

  FIG. 5 shows an example of an optical wavelength multiplexer / demultiplexer. In the figure, the part indicated by the alternate long and short dash line is a schematic enlargement of each part of the optical wavelength multiplexer / demultiplexer. This optical wavelength multiplexer / demultiplexer 500 is of an athermal type using an arrayed waveguide grating (AWG), and includes an input channel waveguide 502, an output channel waveguide 504, a channel waveguide array 506, An input-side slab waveguide 508 and an output-side slab waveguide 510 are provided. In order to realize athermalization, the input side slab waveguide 508 is formed with a deep groove 512 filled with resin. Further, spot size converters 514 and 516 are formed at the input end of the input channel waveguide 502 and the output end of the output channel waveguide 504, respectively, in order to connect to the optical fiber with low loss.

  FIG. 6 is a schematic view of a manufacturing process of the optical wavelength multiplexer / demultiplexer. First, as shown in FIG. 6A, a lower clad 604, a core 606, and an upper clad 608 are sequentially formed on a substrate 602. Next, as shown in FIG. 6B, a resist pattern 610 for forming an athermal groove is formed on the upper clad 608. Then, as shown in FIG. 6C, after forming the groove 612 by reactive ion etching, the resist 610 is removed as shown in FIG. 6D, and as shown in FIG. A silicone resin 614 is filled.

  FIG. 7 shows an optical attenuator as another example. In the figure, the portion indicated by the alternate long and short dash line is a schematic enlargement of each portion of the optical attenuator. This optical attenuator 700 uses an asymmetric Mach-Zehnder interferometer (MZI), and includes two input waveguides 702, two output waveguides 704, and two arm waveguides 706. A phase shifter 710 using a thin film heater is formed on the arm waveguide 706 by being connected by two 3 dB directional couplers 708. Further, on both sides of the arm waveguide, in order to realize low power consumption and low polarization dependence, grooves 712 that block heat propagation and release stress are formed. Further, like the optical wavelength multiplexer / demultiplexer of FIG. 5, in order to connect to an optical fiber with low loss, a spot size converter is provided at the input end of the input waveguide 702 and the output end of the output waveguide 704. 714 and 716 are formed, respectively.

  In recent years, such optical waveguide circuits have been miniaturized by increasing the refractive index of the core to enhance the confinement of light in the waveguide and increasing the density of the waveguide.

Japanese Patent No. 3178565 JP-A-9-61652 J. H. den Besten et al., “Low-Loss, Compact, and Polarization Independent PHASAR Demultiplexer Fabricated by Using a Double-Etch Process,” IEEE Photonics Technology Letters, Vol. 14, No. 1, January 2002

  By the way, in the athermal AWG and MZI shown in FIGS. 5 and 7, a groove or the like must be processed in the manufacturing process. In the athermal AWG, the grooves 514 must be formed so as to cut the core, but in principle, radiation loss occurs in these grooves. In particular, the higher the refractive index of the core, the greater the radiation loss. Therefore, the higher the refractive index of the core, the thinner the groove formed in the waveguide.

  In MZI, in order to reduce power consumption, it is necessary to make the heat insulating grooves 714 formed on both sides of the arm waveguide as close as possible to the waveguide. However, if the groove is too close to the waveguide due to a manufacturing error or the like. , Light oozes and loss increases.

  For the processing of such thin grooves and high-precision grooves, it is better that the etching depth is shallower in order to suppress the dimensional shift accompanying the groove formation. That is, the thinner the upper cladding, the better. This is because if the groove to be etched is deep, the photoresist pattern needs to be thickened, and the thicker the photoresist pattern, the more difficult it is to expose and resolve patterns of fine grooves and high-precision grooves. .

  On the other hand, such an optical waveguide circuit requires a spot size converter for converting the mode field diameter of light confined in the optical waveguide into the mode field diameter of the optical fiber in order to connect to the optical fiber with low loss. As an example, as shown in FIGS. 5 and 7, the spot size converter is formed so that the core width becomes narrower toward the end face of the chip on which the optical circuit is formed. In this spot size converter part, the mode field diameter of the light is enlarged compared to the other parts, so the thickness of the upper cladding must be increased to match the spot size converter part with a larger spot size. I must.

  Thus, in order to process grooves with high accuracy, it is necessary to make the upper clad thinner. However, if the upper clad is made too thin, the spot size converter that expands the spot size of the light will transmit light from the upper clad. There was a problem of seepage and loss increasing.

  The present invention has been made in view of such problems, and an object of the present invention is to provide an optical waveguide circuit that can accurately process a groove and the like while suppressing loss of the optical waveguide circuit to a low level. There is.

In order to achieve such an object, the invention according to claim 1 is directed to an input optical waveguide or an output optical waveguide for connection with an optical fiber, a spot size conversion unit, and other optical waveguides. An optical waveguide circuit having a waveguide portion, wherein the thickness A1 of the upper clad of the other optical waveguide is B1, the thickness of the core of the other optical waveguide is B1, and the maximum of guided light in the other optical waveguide is When the mode field diameter is R1, A1 ≧ (R1−B1) / 2 is satisfied, and the thickness A2 of the upper clad of the spot size conversion unit is B2, and the thickness of the core of the spot size conversion unit is B2. When the maximum mode field diameter of the guided light in the spot size converter is R2, A2 ≧ (R2−B2) / 2 is satisfied, and the thickness A2 of the upper clad of the spot size converter is Other to be thicker than the thickness A1 of the upper cladding of the optical waveguide, i.e., A2> is configured to satisfy the A1, in the region of the upper clad thickness A1, wherein the groove is formed .

The invention according to claim 2 is the optical waveguide circuit according to claim 1 , wherein the region of the thickness A2 of the upper cladding occupies a certain range from the input or output end face of the optical waveguide circuit, A sheet glass is bonded to the region, and an optical fiber is connected to the input optical waveguide or the output waveguide.

According to a third aspect of the present invention, there is provided a method of manufacturing an optical waveguide circuit including a spot size conversion unit, the step of forming a lower cladding on a substrate, and the spot size conversion unit on the lower cladding. Forming a core, and forming an upper clad on the core and the lower clad, wherein the thickness A2 of the upper clad is B2 of the core of the spot size converting portion, and the spot size A step of forming the upper cladding so that A2 ≧ (R2−B2) / 2, where R2 is the maximum mode field diameter of the guided light in the converter, and the upper cladding of the optical waveguide other than the spot size converter the by Tsu steps der etching, the thickness A1 of the upper cladding of the optical waveguide section, the thickness of the core of the optical waveguide portion B1, before When the maximum mode field diameter of the guided light in the optical waveguide section and the R1, A2> A1 ≧ (R1 -B1) / 2 and the step of the upper cladding of the optical waveguide portion is etched so that, of the upper clad thickness And a step of forming a groove in the region A1 .

The invention described in claim 4 is the method for manufacturing the optical waveguide circuit according to claim 3 , further comprising the step of bonding a glass sheet for connecting an optical fiber to the region of the thickness A2 of the upper clad. It is characterized by providing.

  According to the present invention, since the thickness of the upper clad in the portion where the groove of the optical waveguide circuit is formed can be reduced, the processing accuracy of the groove can be increased. Thereby, in an athermal type waveguide, a thin groove can be formed, and loss in the groove can be reduced. Further, in MZI, since a groove with high processing accuracy can be formed, the groove portion can be formed close to the waveguide, and the power consumption can be reduced by improving the heat insulation performance. As a result, it is possible to suppress the deterioration of the loss characteristic which becomes a problem particularly in the waveguide having a high relative refractive index difference Δ, and it is possible to realize a high-performance optical waveguide circuit with low loss.

  Moreover, since the thickness of the upper clad of the spot size converter in which the spot size of the light is enlarged in order to reduce the connection loss with the optical fiber can be increased, the loss in the spot size converter portion is reduced. be able to. Furthermore, if a sufficient adhesion region of the optical fiber connecting plate-attached glass of the optical waveguide circuit is ensured, the reliability of mounting the optical fiber can be sufficiently ensured. In addition, as long as the thick region of the upper clad is limited to the input / output end face of the optical waveguide circuit, the thickness of the resist pattern film for forming the groove pattern does not become uneven.

  Incidentally, in an optical integrated circuit having a spot size converter, a structure is known in which the cladding on the spot size converter is thickened (Patent Document 1). However, in this structure, the target optical waveguide constitutes a laser, and the thickness of the clad layer is adjusted in order to control the spot size. Therefore, in this structure, it is necessary to change the thickness of the upper clad of the spot size converter gently. In the present invention, in order to prevent the waveguide light with the enlarged spot size from seeping out and increasing the loss, it is only necessary to increase the thickness to a certain value or more according to the spot size diameter of the waveguide light. Further, if the range of a certain width is increased from the input / output end of the optical waveguide circuit as well as the upper portion of the spot size converter, a sufficient adhesion area of the glass sheet for connecting optical fibers can be secured.

  Further, as shown in Patent Document 2, a structure is known in which the thickness of the core layer is reduced toward the emission end, and as a result, the upper clad becomes thicker. However, this structure also relates to the laser emission end, and the thickness of the cladding is adjusted to control the spot size. Therefore, this structure also requires the thickness of the upper cladding of the spot size converter to be changed gradually.

  Furthermore, as shown in Non-Patent Document 1, a structure in which etching is performed in two stages is also known. This is because the junction between the slab waveguide and the channel waveguide array of the semiconductor AWG is etched in two stages to reduce the loss generated in this portion. This is a substitute for a stepped step because it is difficult to produce an oblique slope. From a practical point of view, this structure requires that the resist be applied to the uneven surface at the stage where the second etching pattern resist is applied where the step is formed by the first etching. The resist is difficult to be uniform. In the present invention, since the region where the upper cladding is thick may be the end of the input / output waveguide having the spot size converter, there is no need to worry about resist non-uniformity at the time of the second etching.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

  FIG. 1 shows an optical wavelength multiplexer / demultiplexer using an athermal type AWG according to an embodiment of the present invention. In the figure, the part indicated by the alternate long and short dash line is a schematic enlargement of each part of the optical wavelength multiplexer / demultiplexer. The optical wavelength multiplexer / demultiplexer 100 includes an input channel waveguide 102, an output channel waveguide 104, a channel waveguide array 106, an input side slab waveguide 108, and an output side slab waveguide 110. Yes. Further, a triangular groove 112 is formed in the slab waveguide 108, and this groove is filled with a silicone resin to realize an athermal characteristic. Further, spot size converters 114 and 116 are formed at the input end of the input channel waveguide 102 and the output end of the output channel waveguide in order to connect to the optical fiber with low loss. It is known that the groove 112 may be formed in at least one of the input slab waveguide 108, the output slab waveguide 110, and the channel waveguide array 106 according to the design.

  Compared with the conventional optical wavelength multiplexer / demultiplexer 500 shown in FIG. 5, the optical wavelength multiplexer / demultiplexer 100 according to the present invention shown in FIG. 1 has an upper portion of the regions 124 and 126 covering the spot size converters 114 and 116. The clad is thicker than the upper clad in the region 125 where the groove 112 is formed. This is because the mode field diameter of the guided light is enlarged by the spot size converters 114 and 116, so that the upper cladding of the regions 124 and 126 is thickened, while the processing accuracy of the groove 112 is improved, the upper portion of the region 125 is increased. This is a result of thinning the cladding. That is, by increasing the thickness of the upper clad of the spot size converters 114 and 116 in accordance with the mode field diameter of the guided light in the spot size converter, the guided light oozes out of the clad and the loss increases. Can be prevented. On the other hand, by making the upper clad of the groove 112 as thin as possible in accordance with the mode field diameter of the guided light in the core, the groove 112 can be formed thin, and the radiation loss in the groove 112 can be reduced.

In this embodiment, the relative refractive index difference Δ between the core and the clad is 2.5%, the thickness of the lower clad is 15 μm, the thickness of the upper clad in the regions 124 and 126 covering the spot size converter is 15 μm, and the others The thickness of the upper clad in the region 125 was designed to be 5 μm. The spot size converters 114 and 116 are configured such that the waveguide width becomes narrower toward the end face of the optical wavelength multiplexer / demultiplexer 100. However, other configurations may be used as long as the spot size converter has a characteristic that the film thickness of the core does not change or becomes gradually thinner. In such an optical wavelength multiplexer / demultiplexer, in the waveguide of the region 125, the mode field diameter in the vertical direction (light distribution diameter at which the light intensity of light distributed in the core becomes 1 / e ² ) is about 5 μm, and the output The end face of the channel waveguide for use 104 is enlarged to about 10 μm. Therefore, the thickness of the upper clad in the region 125 is 5 μm, and the thickness of the upper clad in the regions 124 and 126 is 15 μm.

If the above-mentioned film thickness design is generalized and described, the thickness A1 of the upper clad is obtained when the thickness of the waveguide core in the region 125 is B1 and the maximum mode field diameter in the vertical direction of the waveguide is R1. Design to satisfy the following equation.
A1 ≧ (R1-B1) / 2

Similarly, the thickness A2 of the upper clad satisfies the following equation when the thickness of the waveguide core in the regions 124 and 126 is B2 and the maximum mode field diameter in the vertical direction of the waveguide is R2. design.
A2 ≧ (R2-B2) / 2> A1

  FIG. 2 is a schematic diagram of a manufacturing process of an optical wavelength multiplexer / demultiplexer according to an embodiment of the present invention. First, as shown in FIG. 2A, a lower clad 204, a core 206, and an upper clad 208 are sequentially formed on a substrate 202. At this time, the thickness A2 of the upper clad is set to 15 μm. Next, as shown in FIG. 2B, a resist pattern 210 is formed on the upper clad 208 in a region covering the spot size converter. Then, as shown in FIG. 2C, etching is performed so that the thickness A1 of the upper clad in the region covering the groove is 5 μm by reactive ion etching. Thereafter, the resist pattern 210 is temporarily removed, and the resist is applied again over the entire surface, and a resist pattern 212 for forming a groove is formed by photolithography (FIG. 2D). Then, as shown in FIG. 2E, a groove 214 is formed by reactive ion etching. The depth of the groove is 13.5 μm in total with the upper cladding 5 μm, the core 3.5 μm, and the lower cladding 5 μm. Finally, the resist 212 is removed (FIG. 2F), and the silicone resin 216 is filled (FIG. 2G).

  In the conventional optical wavelength multiplexer / demultiplexer as shown in FIG. 5, since the thickness of the upper clad is 15 μm, the depth of the groove is 23.5 μm. In the optical wavelength multiplexer / demultiplexer according to the present invention shown in FIG. 1, the groove depth is 13.5 μm, and it can be seen that the etching depth can be greatly reduced. As a result, the minimum width of the processable groove can be reduced from 5 μm to 2.5 μm, and as a result, the radiation loss in the athermal groove can be reduced from 1.5 dB to 0.5 dB.

  FIG. 3 shows a state where an optical fiber is connected to the optical wavelength multiplexer / demultiplexer of FIG. Sheet glass 302 and 304 are bonded on the upper clad of the input channel waveguide and the output channel waveguide to increase the area of the input and output side end faces of the optical wavelength multiplexer / demultiplexer. An input fiber block 312 and an output fiber block 314 are bonded to these end faces, and optical fibers 316 and 318 are connected to the waveguide of the optical wavelength multiplexer / demultiplexer. In practice, the end face that bonds the optical wavelength multiplexer / demultiplexer and the fiber block is polished obliquely to prevent reflection, but is simplified vertically in FIG.

  It is important that the glass plates 302 and 304 are firmly bonded to the upper surface of the optical wavelength multiplexer / demultiplexer for reliability. If the areas of the regions 124 and 126 in FIG. 1 are small, the adhesive strength with the sheet glass 302 and 304 is weakened. The adhesive strength may be effectively increased by, for example, leaving a portion of the upper cladding where the light is not guided in the thick regions 124 and 126 of the upper cladding in a mesh shape. From the viewpoint of adhesive strength, it is desirable that the widths of the regions 124 and 126 are set to be at least half of the width of the sheet glass 302 or 304.

  FIG. 4 shows an optical attenuator using an asymmetric MZI according to an embodiment of the present invention. In the figure, the portion indicated by the alternate long and short dash line is a schematic enlargement of each portion of the optical attenuator. In this optical attenuator 400, two input waveguides 402, two output waveguides 404, and two arm waveguides 406 are connected by two 3 dB directional couplers 408. A phase shifter 410 using a thin film heater is formed on the waveguide 406. Further, on both sides of the arm waveguide, in order to realize low power consumption and low polarization dependency, grooves 412 that block heat propagation and release stress are formed. Further, spot size converters 414 and 416 are formed at the input end of the input waveguide 402 and the output end of the output waveguide 404, respectively, in order to connect to an optical fiber with low loss.

  Compared to the conventional optical attenuator 700 shown in FIG. 7, in the optical attenuator 400 according to the present invention shown in FIG. 4, the upper clad in the regions 424 and 426 covering the spot size converters 414 and 416 has the groove 412. The covering region 425 is thicker than the upper cladding. This is because the mode field diameter of the guided light is enlarged by the spot size converters 414 and 416, so that the upper cladding of the regions 424 and 426 is thickened, while the processing accuracy of the groove 412 is improved, the upper portion of the region 425 is increased. This is a result of thinning the cladding. That is, by increasing the thickness of the upper cladding of the spot size converters 414 and 416 in accordance with the mode field diameter of the guided light in the spot size converter, the guided light oozes out of the cladding and the loss increases. Can be prevented. On the other hand, by making the upper clad of the groove 412 as thin as possible in accordance with the mode field diameter of the guided light in the core, the groove 412 can be formed as close as possible to the core, and the heat insulating effect in the groove 412 is improved. The power consumption in the phase shifter 410 can be reduced.

  In this embodiment, the thickness of the lower clad is 15 μm, the thickness of the core is 3.5 μm, the thickness of the upper clad in the regions 424 and 426 covering the spot size converter is 15 μm, and the upper clad in the other region 425 Was designed to have a thickness of 5 μm. In this case, the depth of the groove 412 is a total of 23.5 μm of the lower cladding 15 μm, the core 3.5 μm, and the upper cladding 5 μm. This is about 70% compared to the depth 33.5 μm (lower cladding 15 μm, core 3.5 μm, upper cladding 15 μm) of the groove 712 of the conventional optical attenuator 700 shown in FIG. As a result, the variation of ± 1 μm pattern in the manufacturing process can be suppressed to ± 0.7 μm.

  The manufacturing process of the optical attenuator 400 is basically the same as that shown in FIG. 2, but a process of forming a thin film heater and wiring in the state shown in FIG. 2C is added. Also in this embodiment, similarly to FIG. 3, the input and output fiber blocks can be bonded, and the optical fiber can be connected to the waveguide of the optical attenuator.

  While the present invention has been described with respect to several embodiments, the embodiments described herein are merely illustrative in view of many possible forms to which the principles of the present invention can be applied. It is not intended to limit the scope of the invention. For example, the present invention can be applied to any waveguide having a relative refractive index difference Δ between the core and the clad. Further, the present invention can be applied to an optical waveguide circuit manufactured by a flame deposition method, a sputtering method, a CVD method or the like without depending on the glass forming method. In addition, when a quartz substrate is used as the substrate, the quartz substrate may also serve as the lower cladding. Furthermore, although the athermal AWG and the asymmetric MZI are taken up as typical optical waveguide circuits, the present invention can be applied to any optical waveguide circuit that has a spot size converter and needs to process a groove or the like. Therefore, the configuration and details of the embodiment exemplified herein can be changed without departing from the spirit of the present invention. Further, the illustrative components and procedures may be changed, supplemented, or changed in order without departing from the spirit of the invention.

It is a figure which shows the optical wavelength multiplexer / demultiplexer by one Example of this invention. It is a figure for demonstrating the manufacturing process of the optical wavelength multiplexer / demultiplexer by one Example of this invention. It is a figure which shows the state which connected the optical fiber to the optical wavelength multiplexer / demultiplexer by one Example of this invention. It is a figure which shows the optical attenuator by one Example of this invention. It is a figure which shows the conventional optical wavelength multiplexer / demultiplexer. It is a figure for demonstrating the manufacturing process of the conventional optical wavelength multiplexer / demultiplexer. It is a figure which shows the conventional optical attenuator.

Explanation of symbols

DESCRIPTION OF SYMBOLS 100 Optical wavelength multiplexer / demultiplexer 102 Input channel waveguide 104 Output channel waveguide 106 Channel waveguide array 108 Input side slab waveguide 110 Output side slab waveguide 112 Groove 114,116 Spot size converter 118 Substrate 124,126 Region where the upper clad is thick 125 Region where the upper clad is thin 202 Substrate 204 Lower clad 206 Core 208 Upper clad 210, 212 Resist pattern 214 Groove 216 Silicone resin 302, 304 Plated glass 312 Input fiber block 314 Output fiber block 316, 318 Light Fiber 400 Optical attenuator 402 Input waveguide 404 Output waveguide 406 Arm waveguide 408 Directional coupler 410 Phase shifter 412 Groove 414, 416 Spot size converter 24,426 Region where the upper cladding is thick 425 Region where the upper cladding is thin 428 Substrate 500 Optical wavelength multiplexer / demultiplexer 502 Input channel waveguide 504 Output channel waveguide 506 Channel waveguide array 508 Input side slab waveguide 510 Output side slab Waveguide 512 Groove 514, 516 Spot size converter 518 Substrate 602 Substrate 604 Lower clad 606 Core 608 Upper clad 610 Resist pattern 612 Groove 614 Silicone resin 700 Optical attenuator 702 Input waveguide 704 Output waveguide 706 Arm waveguide 708 Directional coupler 710 Phase shifter 712 Groove 714, 716 Spot size converter 718 Substrate

Claims (4)

An optical waveguide circuit provided with an input optical waveguide or an output optical waveguide for connection with an optical fiber, a spot size conversion unit, and another optical waveguide unit,
The thickness A1 of the upper clad of the other optical waveguide is expressed by the following equation when the thickness of the core of the other optical waveguide is B1 and the maximum mode field diameter of the guided light in the other optical waveguide is R1. Satisfied,
A1 ≧ (R1-B1) / 2
The thickness A2 of the upper clad of the spot size converter is expressed by the following equation, where B2 is the core thickness of the spot size converter and R2 is the maximum mode field diameter of the guided light in the spot size converter. Satisfied,
A2 ≧ (R2-B2) / 2
The thickness A2 of the upper clad of the spot size converting portion is configured to be larger than the thickness A1 of the upper clad of the other optical waveguides, that is, to satisfy A2> A1, and the thickness of the upper clad An optical waveguide circuit , wherein a groove is formed in a region A1 . The optical waveguide circuit according to claim 1 ,
The region of the thickness A2 of the upper clad occupies a certain range from the input or output end face of the optical waveguide circuit, and a sheet glass is adhered to the region, and an optical fiber is connected to the input optical waveguide or the output waveguide. An optical waveguide circuit configured as described above. A method of manufacturing an optical waveguide circuit including a spot size conversion unit,
Forming a lower cladding on a substrate;
Forming a core with a spot size converter on the lower cladding;
Forming an upper clad on the core and the lower clad, wherein the thickness A2 of the upper clad is B2 of the core of the spot size converter, and the maximum mode of guided light in the spot size converter Forming the upper cladding so that A2 ≧ (R2-B2) / 2 when the field diameter is R2,
Said Tsu step der etching the upper cladding of the optical waveguide portion other than the spot size converting unit, the thickness of the upper cladding of the optical waveguide portion A1 is, the thickness of the core of the optical waveguide portion B1, the optical waveguide Etching the upper clad of the optical waveguide portion so that A2> A1 ≧ (R1-B1) / 2, where R1 is the maximum mode field diameter of the guided light in the portion ;
And a step of forming a groove in a region of the upper clad having a thickness A1 . It is a manufacturing method of the optical waveguide circuit according to claim 3 ,
A method of manufacturing an optical waveguide circuit, further comprising the step of adhering a glass sheet for connecting an optical fiber to a region of thickness A2 of the upper clad.

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