TheCriticalAdiabaticLinearTaperedWaveguideCombinedwith...

Research ArticleThe Critical Adiabatic Linear TaperedWaveguide Combined witha Multimode Waveguide Coupler on an SOI Chip

C L Chiu and Yen-Hsun Liao

Department of Electronic Engineering National Kaohsiung University of Science and Technology No 415 Jiangong RoadKaohsiung 807 Taiwan

Correspondence should be addressed to C L Chiu clchiunkustedutw

Received 10 August 2019 Revised 12 October 2019 Accepted 17 October 2019 Published 11 November 2019

Guest Editor Cheng-Mu Tsai

Copyright copy 2019 C L Chiu and Yen-Hsun Liao +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

A multimode waveguide interference (MMI) coupler is combined with a critical linear tapered waveguide on a silicon-on-insulator (SOI) chip When the TE0 mode is a critical adiabatic mode conversion from a single-mode waveguide to an extremelinear tapered waveguide combined with an MMI this linear tapered waveguide is achieved to the maximum divergence angle(ie the shortest length) +e maximum divergence angle is expressed by θle 2 tanminus 1[(035Wmmi minus Ws)(0172Lmmi)] under a 1times 1MMI combined with this critical linear tapered waveguide +e expression formula is demonstrated by three different widths of a1times 1 MMI of 4 μm8 μm12 μm combined with the critical linear tapered waveguide So the maximum divergence angle isobtained at θ 16deg14deg8deg with respect to this linear tapered waveguide loss of 0022 dB0172 dB0158 dB and this linear tapperlength is reduced by 937929875 than the divergence angle θ 1deg +e output power of a 1times 1 MMI combined with acritical linear tapered waveguide is enhanced at least 15 times under 095 above condition

1 Introduction

In the last few years there have been numerous advances insilicon photonics Photonic devices on a silicon-on-insulator(SOI) chip with high-index contrast have high integrationdensity +e main advantage of the optoelectronic compo-nent on an SOI structure is its good compatibilities [1]Couplers and power dividers in photonic integrated circuits(PICs) are often implemented with multimode interferencecouplers (MMIs) for easy fabrication and broad bandwidth+e SOI platform is an area of interest in integrated optics atpresent and enables a size reduction of PICs+erefore theirCMOS compatibility can provide optoelectronic integrationon a chip in future applications [2] MMIs are based on theexpansion of a fundamental mode of the access waveguideinto multiple modes of the wider width of a multimodewaveguide which interfere as they propagate and formimages of the excitation Ridge waveguides are widely usedin SOI as they offer a single-mode behaviour at micrometrescale [3] MMIs depend on multimode waveguides utilizingbends for higher order mode filtering +ey generally

propagate well and the weak lateral confinement of narrowridge waveguides makes it difficult to achieve high-perfor-mance devices [4]

In 2010 +omson et al proposed a method to achieve areduction of optical loss through the use of linear tapers withinput and output ports +e taper loss is reduced to below1 dB without affecting static extinction [5] In 2012 Shenget al proposed the compact and low-loss MMI couplerfabricated with CMOS technology +is tapered waveguidewith a divergence angle θ of 1deg combined with an MMI isfabricated on SOI with 013 μm CMOS technology to obtainan excess loss of only 006 dB [6] Researchers have adoptedthe widest and longest linear tapered waveguide to becombined with an MMI coupler on an SOI chip in recentdevices +is critical problem will increase manufacturingcosts so it is necessary to design an adiabatic taperedwaveguide

+e losses inherent to a mode propagating waveguidemust be reduced on the cross-sectional boundary betweenthe single-mode waveguide and multimode waveguideBecause a tapered waveguide can change the spot size and

HindawiInternational Journal of OpticsVolume 2019 Article ID 4270612 10 pageshttpsdoiorg10115520194270612

the shape of the optical mode to achieve high couplingefficiency in the cross section boundary [7] a taperedwaveguide is necessary to achieve an adiabatic state [7ndash10]+at is as the TE0 mode from a single-mode waveguide istransmitted to the tapered waveguide the other higher orderTE modes are reduced to excited modes [11ndash17]

In this article we propose an expression formula todesign a terminal linear tapered waveguide to enhance thecoupling efficiency output power of an MMI coupler to twotimes above +e low-loss and maximum divergence anglelinear tapered waveguide combined with a 1times 1 MMI on anSOI chip is achieved to an output power of above 095 +eTE0 mode component ratio is necessary to be above 9785in order to achieve a critical adiabatic mode conversion

2 Device Structure

+e cross section of an SOI structure is shown in Figure 1+e thickness of the upper cladding SiO2 layer is 2 μm andthe Si layer is deposited at a height hco of 220 nm on a 2 -μm-thick buried oxide layer based on a Si substrate +e re-fractive indices of Si and SiO2 are nSi 3475 andnSiO2 1444 respectively +e ridge waveguide has a depthof 222 μm and the effective core refractive index nr of 2509and cladding refractive index nc of 2372 at an operatingwavelength λ0 of 1550 nm in a slab waveguide [18]

MMI couplers have higher tolerance to dimensionalchanges in the fabrication process an easier fabricationprocess than other couplers lower inherent loss large opticalbandwidth and low polarization dependence [19] Multimodewaveguides excite numerous modes depending on their widthand depth +e width of a fixed step index multimodewaveguide Wmmi is generally referred to as Ntimes M MMIcoupler where N and M indicate input and output ports Forhigh-index contrast waveguides the penetration depth is verysmall so thatWeasympWmmi However the effective widthWe cancorrespond to the fundamental mode [18]

We Wmmi +λ0π

1113888 1113889nc

nr1113888 1113889

2σ

n2r minus n

2c1113872 1113873

minus 12 (1)

where λ0 is an operating wavelength and nr and nc are theeffective core and cladding refractive indices respectively +eterm σ 0 represents transverse electric (TE) mode and σ 1is for transverse magnetic (TM)mode Lπ is defined as the beatlength of the two lowest order modes [19] as follows

Lπ π

β0 minus β14nrW

2e

3λ0 (2)

where β0 and β1 are individual zero-order and first-orderpropagation constant +e term nr is the effective core re-fractive index of the slab waveguide from which a 1times 1 MMIcoupler is made We is the effective width of the MMIwaveguide and Lmmi is the exact imaging length [20]

Lmmi 34Lπ (3)

+e geometric shape of a basic 1times 1 MMI coupler isshown in Figure 2(a) A single-mode ridge waveguide with

width Ws of 04 μm length Ls of 100 μm and a depth of222 μm is calculated by the effective core refractive index nrof 2509 and the effective cladding refractive index nc of2372 [17] An MMI width adapted to 102030 times thesingle waveguide of 04 um as an inspecting standard case+e widths of a 1times 1 MMI Wmmi are choice of 4 μm8 μm12 μm respect to the beat lengths Lπ of 457 μm1598 μm3429 μm from equation (2) at an operating wavelength ofλ0 1550 nm +erefore the exact image length of a 1times 1MMI Lmmi achieves 343 μm1198 μm2571 μm re-spectively by equation (3) A linear tapered waveguidecombined with a 1times 1 MMI is shown in Figure 2(b) +einputoutput port of this 1times 1 MMI is a single-modewaveguide linked with a linear tapered waveguide Wt is thewidth and Lt the length of a linear tapered waveguide +edivergence angle θ of a linear tapered waveguide is a taperangle A half angle of the divergence angle is defined by thefollowing equation

tanθ2

1113888 1113889 Wt minus Ws

2Lt (4)

+e simulation analysis utilizes the film mode matchingmethod (FMM) solver in FIMMWAVE software [2122]+e output power of a basic 1times 1 MMI with the exact lengthLmmi is shown in Figure 3 Figure 3(a) is the length Lmmi of a1times 1 MMI scanning the range from 267 μm to 307 μmwitha step of 02 μm at MMI widthWmmi 4 μm +e maximumoutput power is 062 at Lmmi 287 μm Figure 3(b) is thelength Lmmi of a 1times 1 MMI scanning the range from1110 μm to 1150 μm with a step of 02 μm at MMI widthWmmi 8 μm Here the maximum output power is 051 atLmmi 1130 μm Figure 3(c) is the same method at MMIwidth Wmmi 12 μm for Lmmi scanning the range from2552 μm to 2592 μm Maximum output power is 041 atLmmi 2572 μm Accordingly Lmmi at MMI widthWmmi 4 μm8 μm12 μm is 287 μm1130 μm2572 μmrespectively +e device loss of a 1times 1 MMI is 208 dB292 dB387 dB respectively which is very significant

3 Linear Tapered Waveguide Analysis

When the divergence angle of a linear tapered waveguide isset at θ 1deg with a width Wt of 42 μm and a length Lt of2177 μm as an experimental standard result from reference6 this pair of tapered waveguide loss of almost 0004(0018 dB) can be ignored +is linear tapered waveguidewith a divergence angle of 1deg is combined with the threedifferent widths of 1times 1 MMI Wmmi of 4 μm8 μm12 μmwith respect to the exact imaging lengths Lmmi of 287 μm1130 μm2572 μmWhen the ratio ofWtWmmi is increasedfrom 01 to 1 at a step of 005 the range of the output powerincreases from 068 to 1 as shown in Figure 4 +e outputpower of a 1times 1 MMI combined with the linear taperedwaveguide is necessary to be above 095 as the ratio of WtWmmi is set to above 035

Figure 5 shows the effective refractive index neff for eightTE eigenmodes including TE0 TE1 TE2 TE3 TE4 TE5 TE6and TE7 distributed with the width of a linear tapered

2 International Journal of Optics

MMILmmi

Wmmi

Output port

Ws = 04μm

Ls = 100μm

Input portWs

Ls

(a)

Taperedwaveguide

Wmmi

Lmmiθ2

WtWs Ws = 04μm

Ls = 100μmLs Lt Lt

(b)

Figure 2 (a) A basic 1times 1 MMI combined with the inputoutput single-mode waveguide for width Ws 04 μm and length Ls 100 μm(b) A linear tapered waveguide is inserted between the single-mode waveguide and MMI coupler Width Wt length Lt and divergenceangle θ describe the linear tapered waveguide

SiO2 (2μm)

Si substrate

Buried oxide layer (2μm)

Si (220nm)

Claddingnc = 2372

Claddingnc = 2372

Coreneff = 2509

Figure 1 Schematic diagram showing the cross section of a ridge waveguide on an SOI structure

26 27 28 29 30 31

02

03

04

05

06

07

(287 062)P out

Wmmi = 4μm

Lmmi (μm)

MMILmmi

Wmmi

Output portWs = 04μm

Ls = 100μm

Input portWs

Ls

(a)

Figure 3 Continued

International Journal of Optics 3

111 112 113 114 115020025030035040045050055

P out

(1130 051)

Lmmi (μm)

Wmmi = 8μm

(b)

255 256 257 258 259 260030

035

040

P out

(2572 041)

Lmmi (μm)

Wmmi = 12μm

(c)

Figure 3 (a)+emaximum output power is 062 atMMI length Lmmi 287 μmwith a 1times 1MMI widthWmmi 4 μm (b)Maximum outputpower is 051 at Lmmi 1130 μm and Wmmi 8 μm (c) Maximum output power is 041 at Lmmi 2572 μm and Wmmi 12 μm

01 02 03 04 05 06 07 08 09 1003

04

05

06

07

08

09

10

11

12

P out

WtWmmi

Wmmi = 4μm Lmmi = 287μmWmmi = 8μm Lmmi = 1130μmWmmi = 12μm Lmmi = 2572μm

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt

Figure 4 +is linear tapered waveguide with a divergence angle of 1deg is combined with the three different widths of a 1times 1 MMI couplerWmmi of 4 μm8 μm12 μm with respect to the exact imaging lengths Lmmi of 287 μm1130 μm2572 μm When WtWmmi is set at above035 the output power Pout of a 1times 1 MMI coupler combined with a linear tapered waveguide achieves above 095


05 10 15 20 25 30 35 40 4521

22

23

24

25

26

27

28

29

2509

TE7TE6

Effe

ctiv

e ref

ract

ive i

ndex

nef

f

Wt (μm)

Wt = 14μm Wt = 28μm Wt = 42μm

TE0

TE1

TE2

TE3TE4

TE5

Figure 5 +e effective refractive index of the distributed state of eight TE modes with the width of a linear tapered waveguide is rangingfrom 04 μm to 45 μm+e effective refractive index neff of the slab waveguide on an SOI chip is 2509 and the thickness of this linear taperedwaveguide hco is 220 nm at an operating wavelength of 1550 nm

0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

θ = 16deg

TE2 (16 075)TE m

ode c

ompo

nent

ratio

()

θ (degree)

TE0TE1TE2

TE0 (16 9913)

Ls = 100μm Lt

Ws = 04μm Wt = 14μm

(a)

0

20

40

60

80

100

120TE

mod

e com

pone

nt ra

tio (

)

0 5 10 15 20 25 30 35 40 45θ (degree)

TE1TE3TE5

θ = 14deg

TE0 (14 9851)

TE2 (14 134)

Ls = 100μm Lt


TE0TE2TE4

(b)

0

50

100

TE m

ode c

ompo

nent

ratio

()

0 5 10 15 20 25 30 35 40 45θ (degree)

θ = 8deg

TE0 (8 9787)

TE2 (8 197)

Ls = 100μm Lt


TE1TE3TE5TE7TE6

TE0TE2TE4

(c)

Figure 6 +e width of a linear tapered waveguide (a) Wt 14 μm Wmmi 4 μm (b) Wt 28 μm Wmmi 8 μm (c) Wt 42 μmWmmi 12 μm with divergence angle θ of a linear tapered waveguide ranging from 1deg to 45deg +e maximum divergence angle for lineartapered waveguide is achieved at θ 16deg14deg8deg for Wt 14 μm28 μm42 μm respectively


waveguide from 04μm to 45μm As the effective refractiveindex neff of a slab waveguide on an SOI chip is 2509 the eightTE eigenmodes including TE0 to TE7 correspond to the

widths of the linear tapered waveguide Wt +ree differentwidths of 1times 1 MMI Wmmi of 4μm8 μm12μm obtain aminimum width for a linear tapered waveguideWt of 14μm

Input tapered waveguide Output tapered waveguide

zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500

horizontalμm 0 2 4 6 8 10 12

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


Figure 7 A 1times 1MMI in width of 12 μmand in length of 2572 μm is combined with the linear tapered waveguide in widthWt of 42 μmandin length Lt of 2177 μmWhen the location of inputoutput linear tapered waveguide is zLt of 0025050751 the fundamental mode TE0shape of inputoutput port is simulated to change the mode shape size from small to larger mode shape under TE0 adiabatic modeconversion


28μm42μm respectively As the TE0 mode is transmittedfrom a single-mode waveguide with a width of 04μm into alinear tapered waveguide with a width Wt of 14μm TE0 andTE1 are excited +e taper width Wt of 28μm is excited forTE0 TE1 TE2 TE3 and TE4 modes +e taper width Wt of42μm is excited for TE0 TE1 TE2 TE3 TE4 TE5 and TE6modes As the geometric shape of the device is symmetricalstructure the odd modes are suppressed and inexistent

+e single-mode waveguide with width Ws of 04 μm iscombined with the width of the linear tapered waveguideWtof 14 μm28 μm42 μm respectively TE mode componentratio is distributed with the divergence angle of a lineartapered waveguide ranging from 1deg to 45deg as shown inFigure 6 When the even modes of TE2 TE4 and TE6 and theodd modes of TE1 TE3 TE5 and TE7 except TE0 aresuppressed in the linear tapered waveguide the maximumdivergence angle θ of the linear tapered waveguide is 16deg14deg8deg with respect to a 1times 1 MMI with width Wmmi of 4 μm8 μm12 μm +e TE0 mode component ratio obtains indi-vidual 991398519787 So this linear taperedwaveguide achieves the TE0 mode adiabatic mode conver-sion when the TE0 mode component ratio is at least 9787and TE2 mode and the other modes component ratio isbelow 213

For a standard adiabatic mode conversion analysis a1times 1 MMI in width of 12 μm and in length of 2572 μm iscombined with the divergence angle θ 1deg of linear taperedwaveguide in width Wt of 42 μm and in length Lt of

2177 μm When the location of inputoutput linear taperedwaveguide is zLt of 0025050751 the fundamentalmode TE0 shape of inputoutput port is simulated to changethe mode shape size from smaller to larger mode shapeunder TE0 adiabatic mode conversion as shown in Figure 7+e coupling efficiency of this device between the single-mode waveguide and the multimode waveguide is enhancedfrom 041 to 095

+e 1times 1 MMI with width Wmmi of 4 μm8 μm12 μm iscombined with the width of the linear tapered waveguideWtof 14 μm28 μm42 μm respectively as the input andoutput port with the divergence angle θ of a linear taperedwaveguide scanning the range from 1deg to 45deg at a step of 1deg+emaximum divergence angle θ is achieved at 16deg14deg8deg re-spectively under the condition of output power of a 1times 1MMIcombined with a linear tapered waveguide of at least 095 asshown in Figure 8 Figure 9 shows that the spectral responsesare insensitivity for the wavelength from 1546 to 1554nmwiththe step of 1 nm under the condition of this linear taperedwaveguide with a maximum divergence angle θ of 16deg14deg8degcombined with the three different widths of a 1times 1 MMI of4μm8 μm12μm respectively +e output power of threedifferent widths of a 1times 1 MMI linked with the maximumdivergence angle θ of 16deg14deg8deg of a linear taperedwaveguide is095 when the ratio of WtWmmi is equal to 035 +ree dif-ferent divergence angles θ of the linear tapered waveguide of16deg14deg8deg with respect to three different widths of a 1times 1MMIwidth Wmmi of 4μm8μm12 μm are taken into equation (4)

0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out

Wmmi = 4μm Lmmi = 287μm Wt = 14μmWmmi = 8μm Lmmi = 1130μm Wt = 28μmWmmi = 12 μm Lmmi = 2572μm Wt = 42μm

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt

Figure 8 WhenWtWmmi is set at 035 the 1times 1 MMI coupler withWmmi 4 μm8 μm12 μm achieves a minimum width of linear taperedwaveguide atWt 14 μm28 μm42 μm respectively As the divergence angle θ of a linear tapered waveguide is scanning the range from 1degto 45deg at a step of 1deg a maximum divergence angle θ 16deg14deg8deg is obtained under the constraint of Pout ge 095 respectively


+e length of a linear tapered waveguide Lt is calculated by36μm98μm272μm respectively +e ratio of the length ofa linear tapered waveguide to the length of a 1times 1 MMI isexpressed as LtLmmi≧ 0086

+e expressions of equations (5) and (6) are demon-strated under three different widths of a 1times 1 MMI couplercombined with a designed linear tapered waveguide

Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)

where Wt is the width of the linear tapered waveguide Lt isthe length of the linear tapered waveguide and Wmmi is thewidth of a 1times 1MMI and Lmmi of the exact imaging length ofa 1times 1 MMI When the width of the single-mode waveguideWs of 04 μm and equations (5) and (6) are taken intoequation (4) the maximum divergence angle θ is expressedas equation (7)

θ le 2 tanminus 1 035Wmmi minus Ws

0172Lmmi1113888 1113889 (7)

Comparison of basic 1times 1 MMI device loss with a 1times 1MMI combined with a linear tapered waveguide device lossis shown in Tables 1 and 2 When 1times 1 MMI with widthWmmi of 4 μm8 μm12 μm is combined with a maximumdivergence angle θ= 16deg14deg8deg in a linear tapered waveguidewith a width Wt of 14 μm28 μm42 μm and length Lt of36 μm98 μm272 μm the loss of this linear taperedwaveguide is 0022 dB0172 dB0158 dB +e length of amaximum divergence angle θ= 16deg14deg8deg in this linear ta-pered waveguide is reduced to 937929875 than thelength of the divergence angle θ= 1deg combined a 1times 1 MMIwith width of 4 μm8 μm12 μm+e output power of a 1times 1MMI combined with the maximum divergence angle of alinear tapered waveguide is 095 (022 dB) A 1times 1 MMIdevice loss with a linear tapered waveguide reduces 186 dB270 dB365 dB than a 1times 1MMI device loss without a linear

1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt

Figure 9+e spectral responses are insensitivity for the wavelength from 1546 to 1554 nm with the step of 1 nm under the condition of thislinear tapered waveguide with a maximum divergence angle θ of 16deg14deg8deg combined with the three different widths of a 1times 1MMI of 4 μm8 μm12 μm respectively

Table 1 A basic 1times 1 MMI device loss

W mmi (μm) L mmi (μm) Pout (device loss (dB))4 287 062(208 dB)8 1130 051(292 dB)12 2572 041(387 dB)


tapered waveguide +is device loss represents a significantreduction

4 Conclusion

A 1 times 1 MMI is combined with a symmetrical linear ta-pered waveguide on an SOI chip When TE0 mode from asingle-mode waveguide is transmitted to this criticallinear tapered waveguide linked with a 1 times 1 MMI the TE0mode component ratio is necessary to be at least 9787and the TE2 mode and the other modesrsquo component ratiosare to be below 213 So the TE0 mode presents a criticaladiabatic mode conversion +e designed linear taperedwaveguide is achieved to the shortest length and themaximum divergence angle

Under the condition of a 1times 1 MMI coupler combinedwith the designed linear tapered waveguide the maximumdivergence angle is demonstrated by θle 2 tanminus 1

[(035Wmmi minus Ws)(0172 Lmmi)] When the width of a 1times 1MMI Wmmi is 4 μm8 μm12 μm with respect to the lengthLmmi of 287 μm1130 μm2572 μm the maximum di-vergence angle θ is achieved to 16deg14deg8deg respectively

A 1times 1MMIwidthWmmi of 4 μm8 μm12 μmcombinedwith a maximum divergence angle θ= 16deg14deg8deg linear ta-pered waveguide to a 1times 1 MMI without linear taperedwaveguide +e simulation result shows that the device lossis reduced by 186 dB270 dB365 dB respectively withrespect to an extreme linear tapered waveguide loss of0022 dB0172 dB0158 dB +e length of a maximum di-vergence angle θ= 16deg14deg8deg linear tapered waveguide isreduced to 937929875 than the length of the di-vergence angle θ= 1deg linear tapered waveguide combinedwith a 1times 1 MMI with width of 4 μm8 μm12 μm +eoutput power of a 1times 1 MMI combined with a critical lineartapered waveguide is at least 095 which enhanced thecoupling efficiency by 15 times

Data Availability

+e data used to support the findings of this study are in-cluded within the article files

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported in part by the Ministry of Scienceand Technology Taiwan Republic of China under the grantno MOST 108-2221-E-992-080

References

[1] R Soref ldquo+e past present and future of silicon photonicsrdquoIEEE Journal of Selected Topics in Quantum Electronicsvol 12 no 6 pp 1678ndash1687 2006

[2] M Lipson ldquoGuiding modulating and emitting light on sil-icon-challenges and opportunitiesrdquo Journal of LightwaveTechnology vol 23 no 12 pp 4222ndash4238 2005

[3] K Kruse and C T Middlebrook ldquoPolymer taper bridge forsilicon waveguide to single mode waveguide couplingrdquo OpticsCommunications vol 362 pp 87ndash95 2016

[4] J Guo and Y Zhao ldquoAnalysis of mode hybridization in ta-pered waveguidesrdquo IEEE Photonics Technology Letters vol 27no 23 pp 2441ndash2444 2015

[5] D J +omson Y Hu G T Reed and J-M Fedeli ldquoLow lossMMI couplers for high performance MZI modulatorsrdquo IEEEPhotonics Technology Letters vol 22 no 20 pp 1485ndash14872010

[6] Z Sheng Z Zhiqi Wang C Chao Qiu et al ldquoA compact andlow-loss MMI coupler fabricated with CMOS technologyrdquoIEEE Photonics Journal vol 4 no 6 pp 2272ndash2277 2012

[7] P Sethi A Haldar and S K Selvaraja ldquoUltra-compact low-loss broadband waveguide taper in silicon-on-insulatorrdquoOptics Express vol 25 no 9 pp 10196ndash10203 2017

[8] Y Liu W Sun H Xie et al ldquoAdiabatic and ultra-compactwaveguide tapers based on digital metamaterialsrdquo IEEEJournal of Selected Topics in Quantum Electronics vol 25no 3 pp 1ndash6 2018

[9] J Zhang J Yang H Xin J Huang D Chen and Z ZhaojianldquoUltrashort and efficient adiabatic waveguide taper based onthin flat focusing lensesrdquo Optics Express vol 25 no 17pp 19894ndash19903 2017

[10] Y Fu T Ye W Tang and T Chu ldquoEfficient adiabatic silicon-on-insulator waveguide taperrdquo Photonics Research vol 2no 3 pp A41ndashA44 2014

[11] J Wang M Qi Y Xuan et al ldquoProposal for fabrication-tolerant SOI polarization splitter-rotator based on cascadedMMI couplers and an assisted bi-level taperrdquo Optics Expressvol 22 no 23 pp 27869ndash27879 2014

[12] Y Zhang S Yang A E-J Lim et al ldquoA CMOS-compatiblelow-loss and low-crosstalk silicon waveguide crossingrdquo IEEEPhotonics Technology Letters vol 25 no 5 pp 422ndash425 2013

[13] D Dai Y Tang and J E Bowers ldquoMode conversion in ta-pered submicron silicon ridge optical waveguidesrdquo OpticsExpress vol 20 no 12 pp 13425ndash13439 2012

[14] L He Y He A Pomerene et al ldquoUltrathin silicon-on-in-sulator grating couplersrdquo IEEE Photonics Technology Lettersvol 24 no 24 pp 2247ndash2249 2012

[15] C-H Chen and C-H Chiu ldquoTaper-integrated multimode-interference based waveguide crossing designrdquo IEEE Journalof Quantum Electronics vol 46 no 11 pp 1656ndash1661 2010

[16] W Bogaerts P Dumon D V+ourhout and R Baets ldquoLow-loss low-cross-talk crossings for silicon-on-insulator nano-photonic waveguidesrdquo Optics Letters vol 32 no 19pp 2801ndash2803 2007

Table 2 A 1times 1 MMI combined with a linear tapered waveguide device loss

Pout (Device loss (dB)) W t (μm) Max divergence angle (degree) Linear tapered waveguide loss (dB)095 (022 dB) 14 16deg 0022095 (022 dB) 28 14deg 0172095 (022 dB) 42 8deg 0158


[17] J J Wu B R Shi and M Kong ldquoExponentially taperedmultimode interference couplersrdquo Chinese Optics Lettersvol 4 no 3 pp 167ndash169 2006

[18] P K Bhattacharya Semiconductor Optoelectronic DevicesPrentice-Hall Englewood Cliffs NJ USA 1998

[19] L B Soldano and E C M Pennings ldquoOptical multi-modeinterference devices based on self-imaging principles andapplicationsrdquo Journal of Lightwave Technology vol 13 no 4pp 615ndash627 1995

[20] M Bachmann P A Besse and H Melchior ldquoOverlapping-image multimode interference couplers with a reducednumber of self-images for uniform and nonuniform powersplittingrdquo Applied Optics vol 34 no 30 pp 6898ndash6910 1995

[21] A S Sudbo ldquoFilm mode matching a versatile numericalmethod for vector mode field calculations in dielectricwaveguidesrdquo Pure and Applied Optics Journal of the EuropeanOptical Society Part A vol 2 no 3 pp 211ndash233 1993

[22] Integrated Optics Software FIMMWAVE 52 Photon DesignOxford UK 2015 httpwwwphotondcom


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the shape of the optical mode to achieve high couplingefficiency in the cross section boundary [7] a taperedwaveguide is necessary to achieve an adiabatic state [7ndash10]+at is as the TE0 mode from a single-mode waveguide istransmitted to the tapered waveguide the other higher orderTE modes are reduced to excited modes [11ndash17]

In this article we propose an expression formula todesign a terminal linear tapered waveguide to enhance thecoupling efficiency output power of an MMI coupler to twotimes above +e low-loss and maximum divergence anglelinear tapered waveguide combined with a 1times 1 MMI on anSOI chip is achieved to an output power of above 095 +eTE0 mode component ratio is necessary to be above 9785in order to achieve a critical adiabatic mode conversion

2 Device Structure

+e cross section of an SOI structure is shown in Figure 1+e thickness of the upper cladding SiO2 layer is 2 μm andthe Si layer is deposited at a height hco of 220 nm on a 2 -μm-thick buried oxide layer based on a Si substrate +e re-fractive indices of Si and SiO2 are nSi 3475 andnSiO2 1444 respectively +e ridge waveguide has a depthof 222 μm and the effective core refractive index nr of 2509and cladding refractive index nc of 2372 at an operatingwavelength λ0 of 1550 nm in a slab waveguide [18]

MMI couplers have higher tolerance to dimensionalchanges in the fabrication process an easier fabricationprocess than other couplers lower inherent loss large opticalbandwidth and low polarization dependence [19] Multimodewaveguides excite numerous modes depending on their widthand depth +e width of a fixed step index multimodewaveguide Wmmi is generally referred to as Ntimes M MMIcoupler where N and M indicate input and output ports Forhigh-index contrast waveguides the penetration depth is verysmall so thatWeasympWmmi However the effective widthWe cancorrespond to the fundamental mode [18]

We Wmmi +λ0π

1113888 1113889nc

nr1113888 1113889

2σ

n2r minus n

2c1113872 1113873

minus 12 (1)

where λ0 is an operating wavelength and nr and nc are theeffective core and cladding refractive indices respectively +eterm σ 0 represents transverse electric (TE) mode and σ 1is for transverse magnetic (TM)mode Lπ is defined as the beatlength of the two lowest order modes [19] as follows

Lπ π

β0 minus β14nrW

2e

3λ0 (2)

where β0 and β1 are individual zero-order and first-orderpropagation constant +e term nr is the effective core re-fractive index of the slab waveguide from which a 1times 1 MMIcoupler is made We is the effective width of the MMIwaveguide and Lmmi is the exact imaging length [20]

Lmmi 34Lπ (3)

+e geometric shape of a basic 1times 1 MMI coupler isshown in Figure 2(a) A single-mode ridge waveguide with

width Ws of 04 μm length Ls of 100 μm and a depth of222 μm is calculated by the effective core refractive index nrof 2509 and the effective cladding refractive index nc of2372 [17] An MMI width adapted to 102030 times thesingle waveguide of 04 um as an inspecting standard case+e widths of a 1times 1 MMI Wmmi are choice of 4 μm8 μm12 μm respect to the beat lengths Lπ of 457 μm1598 μm3429 μm from equation (2) at an operating wavelength ofλ0 1550 nm +erefore the exact image length of a 1times 1MMI Lmmi achieves 343 μm1198 μm2571 μm re-spectively by equation (3) A linear tapered waveguidecombined with a 1times 1 MMI is shown in Figure 2(b) +einputoutput port of this 1times 1 MMI is a single-modewaveguide linked with a linear tapered waveguide Wt is thewidth and Lt the length of a linear tapered waveguide +edivergence angle θ of a linear tapered waveguide is a taperangle A half angle of the divergence angle is defined by thefollowing equation

tanθ2

1113888 1113889 Wt minus Ws

2Lt (4)

+e simulation analysis utilizes the film mode matchingmethod (FMM) solver in FIMMWAVE software [2122]+e output power of a basic 1times 1 MMI with the exact lengthLmmi is shown in Figure 3 Figure 3(a) is the length Lmmi of a1times 1 MMI scanning the range from 267 μm to 307 μmwitha step of 02 μm at MMI widthWmmi 4 μm +e maximumoutput power is 062 at Lmmi 287 μm Figure 3(b) is thelength Lmmi of a 1times 1 MMI scanning the range from1110 μm to 1150 μm with a step of 02 μm at MMI widthWmmi 8 μm Here the maximum output power is 051 atLmmi 1130 μm Figure 3(c) is the same method at MMIwidth Wmmi 12 μm for Lmmi scanning the range from2552 μm to 2592 μm Maximum output power is 041 atLmmi 2572 μm Accordingly Lmmi at MMI widthWmmi 4 μm8 μm12 μm is 287 μm1130 μm2572 μmrespectively +e device loss of a 1times 1 MMI is 208 dB292 dB387 dB respectively which is very significant

3 Linear Tapered Waveguide Analysis

When the divergence angle of a linear tapered waveguide isset at θ 1deg with a width Wt of 42 μm and a length Lt of2177 μm as an experimental standard result from reference6 this pair of tapered waveguide loss of almost 0004(0018 dB) can be ignored +is linear tapered waveguidewith a divergence angle of 1deg is combined with the threedifferent widths of 1times 1 MMI Wmmi of 4 μm8 μm12 μmwith respect to the exact imaging lengths Lmmi of 287 μm1130 μm2572 μmWhen the ratio ofWtWmmi is increasedfrom 01 to 1 at a step of 005 the range of the output powerincreases from 068 to 1 as shown in Figure 4 +e outputpower of a 1times 1 MMI combined with the linear taperedwaveguide is necessary to be above 095 as the ratio of WtWmmi is set to above 035

Figure 5 shows the effective refractive index neff for eightTE eigenmodes including TE0 TE1 TE2 TE3 TE4 TE5 TE6and TE7 distributed with the width of a linear tapered


MMILmmi

Wmmi

Output port

Ws = 04μm

Ls = 100μm

Input portWs

Ls

(a)

Taperedwaveguide

Wmmi

Lmmiθ2

WtWs Ws = 04μm

Ls = 100μmLs Lt Lt

(b)


SiO2 (2μm)

Si substrate


Si (220nm)

Claddingnc = 2372

Claddingnc = 2372

Coreneff = 2509


26 27 28 29 30 31

02

03

04

05

06

07

(287 062)P out

Wmmi = 4μm

Lmmi (μm)

MMILmmi

Wmmi


Ls = 100μm

Input portWs

Ls

(a)

Figure 3 Continued


111 112 113 114 115020025030035040045050055

P out

(1130 051)

Lmmi (μm)

Wmmi = 8μm

(b)

255 256 257 258 259 260030

035

040

P out

(2572 041)

Lmmi (μm)

Wmmi = 12μm

(c)


01 02 03 04 05 06 07 08 09 1003

04

05

06

07

08

09

10

11

12

P out

WtWmmi


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt



05 10 15 20 25 30 35 40 4521

22

23

24

25

26

27

28

29

2509

TE7TE6

Effe

ctiv

e ref

ract

ive i

ndex

nef

f

Wt (μm)


TE0

TE1

TE2

TE3TE4

TE5


0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

θ = 16deg

TE2 (16 075)TE m

ode c

ompo

nent

ratio

()

θ (degree)

TE0TE1TE2

TE0 (16 9913)

Ls = 100μm Lt


(a)

0

20

40

60

80

100

120TE

mod

e com

pone

nt ra

tio (

)

0 5 10 15 20 25 30 35 40 45θ (degree)

TE1TE3TE5

θ = 14deg

TE0 (14 9851)

TE2 (14 134)

Ls = 100μm Lt


TE0TE2TE4

(b)

0

50

100

TE m

ode c

ompo

nent

ratio

()

0 5 10 15 20 25 30 35 40 45θ (degree)

θ = 8deg

TE0 (8 9787)

TE2 (8 197)

Ls = 100μm Lt


TE1TE3TE5TE7TE6

TE0TE2TE4

(c)






zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500









0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





MMILmmi

Wmmi

Output port

Ws = 04μm

Ls = 100μm

Input portWs

Ls

(a)

Taperedwaveguide

Wmmi

Lmmiθ2

WtWs Ws = 04μm

Ls = 100μmLs Lt Lt

(b)


SiO2 (2μm)

Si substrate


Si (220nm)

Claddingnc = 2372

Claddingnc = 2372

Coreneff = 2509


26 27 28 29 30 31

02

03

04

05

06

07

(287 062)P out

Wmmi = 4μm

Lmmi (μm)

MMILmmi

Wmmi


Ls = 100μm

Input portWs

Ls

(a)

Figure 3 Continued


111 112 113 114 115020025030035040045050055

P out

(1130 051)

Lmmi (μm)

Wmmi = 8μm

(b)

255 256 257 258 259 260030

035

040

P out

(2572 041)

Lmmi (μm)

Wmmi = 12μm

(c)


01 02 03 04 05 06 07 08 09 1003

04

05

06

07

08

09

10

11

12

P out

WtWmmi


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt



05 10 15 20 25 30 35 40 4521

22

23

24

25

26

27

28

29

2509

TE7TE6

Effe

ctiv

e ref

ract

ive i

ndex

nef

f

Wt (μm)


TE0

TE1

TE2

TE3TE4

TE5


0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

θ = 16deg

TE2 (16 075)TE m

ode c

ompo

nent

ratio

()

θ (degree)

TE0TE1TE2

TE0 (16 9913)

Ls = 100μm Lt


(a)

0

20

40

60

80

100

120TE

mod

e com

pone

nt ra

tio (

)

0 5 10 15 20 25 30 35 40 45θ (degree)

TE1TE3TE5

θ = 14deg

TE0 (14 9851)

TE2 (14 134)

Ls = 100μm Lt


TE0TE2TE4

(b)

0

50

100

TE m

ode c

ompo

nent

ratio

()

0 5 10 15 20 25 30 35 40 45θ (degree)

θ = 8deg

TE0 (8 9787)

TE2 (8 197)

Ls = 100μm Lt


TE1TE3TE5TE7TE6

TE0TE2TE4

(c)






zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500









0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





111 112 113 114 115020025030035040045050055

P out

(1130 051)

Lmmi (μm)

Wmmi = 8μm

(b)

255 256 257 258 259 260030

035

040

P out

(2572 041)

Lmmi (μm)

Wmmi = 12μm

(c)


01 02 03 04 05 06 07 08 09 1003

04

05

06

07

08

09

10

11

12

P out

WtWmmi


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt



05 10 15 20 25 30 35 40 4521

22

23

24

25

26

27

28

29

2509

TE7TE6

Effe

ctiv

e ref

ract

ive i

ndex

nef

f

Wt (μm)


TE0

TE1

TE2

TE3TE4

TE5


0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

θ = 16deg

TE2 (16 075)TE m

ode c

ompo

nent

ratio

()

θ (degree)

TE0TE1TE2

TE0 (16 9913)

Ls = 100μm Lt


(a)

0

20

40

60

80

100

120TE

mod

e com

pone

nt ra

tio (

)

0 5 10 15 20 25 30 35 40 45θ (degree)

TE1TE3TE5

θ = 14deg

TE0 (14 9851)

TE2 (14 134)

Ls = 100μm Lt


TE0TE2TE4

(b)

0

50

100

TE m

ode c

ompo

nent

ratio

()

0 5 10 15 20 25 30 35 40 45θ (degree)

θ = 8deg

TE0 (8 9787)

TE2 (8 197)

Ls = 100μm Lt


TE1TE3TE5TE7TE6

TE0TE2TE4

(c)






zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500









0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





05 10 15 20 25 30 35 40 4521

22

23

24

25

26

27

28

29

2509

TE7TE6

Effe

ctiv

e ref

ract

ive i

ndex

nef

f

Wt (μm)


TE0

TE1

TE2

TE3TE4

TE5


0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

120

θ = 16deg

TE2 (16 075)TE m

ode c

ompo

nent

ratio

()

θ (degree)

TE0TE1TE2

TE0 (16 9913)

Ls = 100μm Lt


(a)

0

20

40

60

80

100

120TE

mod

e com

pone

nt ra

tio (

)

0 5 10 15 20 25 30 35 40 45θ (degree)

TE1TE3TE5

θ = 14deg

TE0 (14 9851)

TE2 (14 134)

Ls = 100μm Lt


TE0TE2TE4

(b)

0

50

100

TE m

ode c

ompo

nent

ratio

()

0 5 10 15 20 25 30 35 40 45θ (degree)

θ = 8deg

TE0 (8 9787)

TE2 (8 197)

Ls = 100μm Lt


TE1TE3TE5TE7TE6

TE0TE2TE4

(c)






zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500









0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








zLt = 0

zLt = 025

zLt = 05

zLt = 075

zLt = 1

z = 0 z = 0z = Lt z = Lt

vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500


vert

ical

μm

403530252015100500









0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










0 5 10 15 20 25 30 35 40 4505

06

07

08

09

10

11

12

(16 095)

θ (degree)

(14 095)

(8 095)

P out


Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt





Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







Wt ge 035Wmmi (5)

Lt ge 0086Lmmi (6)



0172Lmmi1113888 1113889 (7)


1546 1547 1548 1549 1550 1551 1552 1553 155405

06

07

08

09

10

11

12

P out

Wavelength (nm)


095

Taperedwaveguide

Wmmi

Lmmiθ2

Wt

Ws = 04μm

Ls = 100μmLt






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






4 Conclusion





Data Availability




Acknowledgments


References






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery















Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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