Analysis of silicon nitride partial Euler waveguide bends
Florian Vogelbacher, Stefan Nevlacsil, Martin Sagmeister, Jochen Kraft, Karl Unterrainer, Rainer Hainberger
AAnalysis of silicon nitride partial Eulerwaveguide bends F LORIAN V OGELBACHER , S TEFAN N EVLACSIL , M ARTIN S AGMEISTER , J OCHEN K RAFT , K ARL U NTERRAINER , AND R AINER H AINBERGER AIT Austrian Institute of Technology GmbH, Center for Health & Bioresources, Giefinggasse 4,1210 Vienna, Austria TU Wien, Photonics Institute, Gusshausstraße 27-29, 1040 Vienna, Austria ams AG, Tobelbader Straße 30, 8141 Premstätten, Austria * fl[email protected] Abstract:
In this work we present a detailed analysis of individual loss mechanisms in siliconnitride partial Euler bends at a wavelength of 850 nm. This structure optimizes the transmissionthrough small radii optical waveguide bends. The partial Euler bend geometry balances lossesarising from the transition from the straight to the bend waveguide mode, and radiative losses ofthe bend waveguide mode. Numerical analyses are presented for 45-degree bends commonlyemployed in S-bend configurations to create lateral offsets, as well as 90- and 180-degree bends.Additionally, 90-degree partial Euler bends were fabricated on a silicon nitride photonic platformto experimentally complement the theoretical findings. The optimized waveguide bends allowfor a reduced effective radius without increasing the total bend loss and, thus, enable a highercomponent density in photonic integrated circuits. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic integrated circuits (PICs) play an important role in data- and telecommunication [1, 2]and are attracting increasing interest in other fields of applications such as astronomy [3, 4],biosensing [5, 6], medical diagnostics [7, 8], and quantum photonics [9]. These applicationsbenefit from the strongly confined light, stable operation, and the small footprint compared tobulk optics. However, the compactness of a PIC is significantly limited by losses arising fromwaveguide bends used in routing of optical signals. Therefore, the optimization of bends is ofpivotal importance in the design of high-density PICs or devices requiring ultra-low losses.Different approaches to reduce total bend losses have been proposed in literature. For example,a lateral offset between the incoming straight waveguide and the bend can reduce the modemismatch between these two sections [10]. However, this technique relies on a high fabricationresolution, which is not always attainable by standard photolithography methods. Fabricationlimitations are relaxed when a smooth transition between the straight waveguide and the bendis introduced. Ideally, such a transition allows to adiabatically convert the mode profile of astraight waveguide to that of a bent section. In the matched bend approach the length of thebend is optimized to minimize the higher order leaky modes at the output of the bend [11].Alternatively, a variation of the waveguide width, a multi-step patterning with grooves along thebend or subwavelength gratings on the waveguide can alter the mode confinement and reducelosses in the bend [12–14]. The implementation of groves and subwavelength gratings requiresadditional fabrication steps and increases costs. In many applications a bend loss reductionby an optimized geometry alone is therefore favorable. Various bend geometries have beenproposed, e.g. based on spline curves [12], trigonometric functions [15], or deduced from atopology optimization [16–19]. The variational method [20, 21] relies on analytic bend lossmodels for deriving an optimized bend geometry. However, this approach is inherently limited a r X i v : . [ phy s i c s . op ti c s ] O c t y the physical validity of the applied bend loss model and, thus, carries the risk of omitting orinadequately describing critical loss mechanisms.A bend geometry of particular interest is based on the Euler spiral, also known as a clothoid.Euler spirals are widely employed in civil engineering as transition curves because of thecontinuous change in curvature and straightforward implementation [22, 23]. An Euler spiralis defined through a linear increase of the bend curvature along the path length, resulting in acontinuous transition from the straight waveguide to the waveguide bend. This transition reducesthe excitation of higher order modes [24]. Optical waveguide bends based on Euler spirals havebeen successfully demonstrated for single mode and multimode optical waveguides [24, 25].The combination of an Euler bend with a section of constant curvature has been presented byFujisawa et al. for a silicon photonic platform in [26]. This combination balances two major losscomponents of a waveguide bend, namely the transition losses arising from a changing curvature,and radiative losses inherent to a bend waveguide mode.In this work, we present a detailed numerical analysis of the partial Euler bend geometry. Areformulated description to construct the partial Euler bend geometry compared to the work ofFujisawa et al. [26] facilitates the implementation. This formulation allows to implement arbitraryangled bends <
2. Theory
In this section, the geometric properties of an Euler spiral are presented and subsequently used forthe construction of a partial Euler bend consisting of a section with linearly increasing curvatureand a section of constant curvature. Figure 1(a) exemplarily depicts the geometry of a 90°partial Euler bend and indicates the parameters used to describe a partial Euler bend. The bendparameter p relates to the portion of the bend having a linearly increasing curvature. The inputand output of a partial Euler bend with effective radius R eff coincide with that of a circular bendof the same radius. The tangent lines of the input and output are preserved. The bend geometryis expressed in Cartesian coordinates to enable a straightforward implementation in a photonicdesign software. In Cartesian coordinates the Euler spiral can be constructed through the scaled Fresnel integrals[32] x ( s ) = ∫ s cos (cid:32) t R (cid:33) dt (1a) y ( s ) = ∫ s sin (cid:32) t R (cid:33) dt (1b)where s denotes the path length and R is a free to choose radius. For example, this parametercan be set to 1 /√ κ y R min η (x p ,y p ) α /2 p = 0 p = 0.5 R eff R eff (1- p ) α (a) (b) (c) (d) Fig. 1. (a) Geometry of a 90° partial Euler bend with bend parameter p = . p influences the angle at which the transition between thesection of linearly increasing curvature (blue) to the section with constant curvature (lightred) occurs. For p = R min . (b) 45°, (c) 90°, and (d) 180° bend geometries for p = . increases linearly with the path length, i.e., κ ( s ) = sR (2)and the angle α is reached at a path length s ( α ) = R √ α. (3)The curvature at an angle α is accordingly given by κ ( α ) = √ α R . (4)The integrals in Eq. (1a) and Eq. (1b) cannot be solved analytically, but numerical integration ora series expansion can be used to solve these with suitable precision [32–35]. The presented partial Euler bend of total bend angle α is designed symmetrically around α /
2. Inthe following the first half of the bend is described. The first section up to an angle p α / α /
2. Thesecond half is readily obtained from a mirroring operation. To construct a partial Euler bendwith an effective radius R eff , the functions describing the bend need to be rescaled. The rescalingfactor depends on the bend angle α , the effective radius R eff , and the bend parameter p .In a first step, the coordinates ( x p , y p ) of the end of the Euler section are calculated fromEq. (1a) and Eq. (1b), i.e., x p = x ( s p ) and y p = y ( s p ) , where s p = R √ p α denotes the pathlength to reach the angle p α /
2. The radius of curvature R p at ( x p , y p ) is given from Eq. (2), andresults in R p = κ ( s p ) = R √ p α . (5)he offset to translate the start of the circular section to the end of the Euler section is calculatedwith ∆ x = x p − R p sin ( p α / ) (6a) ∆ y = y p − R p [ − cos ( p α / )] . (6b)The total bend length is the sum of the path lengths of two Euler sections and the circular sectionwith radius R p resulting in s = s p + R p α ( − p ) . (7)Combining the results, the functions describing the first half of the unscaled partial Euler bendfor angles smaller than α / x bend ( s ) = (cid:40) x ( s ) for 0 ≤ s ≤ s p R p sin (cid:16) s − s p R p + p α (cid:17) + ∆ x for s p < s ≤ s / y bend ( s ) = (cid:40) y ( s ) for 0 ≤ s ≤ s p R p (cid:104) − cos (cid:16) s − s p R p + p α (cid:17)(cid:105) + ∆ y for s p < s ≤ s / η to achieve an effective radius R eff tobe η = R eff y bend ( s / ) + x bend ( s / ) / tan ( α / ) . (9)The equations describing the rescaled bend can be written as˜ x bend ( s ) = η · x bend ( s / η ) (10a)˜ y bend ( s ) = η · y bend ( s / η ) . (10b)with the minimum radius of curvature given by R min = η · R p . (11)The path length to construct the first half of the rescaled bend ranges from0 ≤ s ≤ η s . (12)The geometric construction allows for bend angles < η in Eq. (9) depends on the parameter p of the partial Euler bend as well ason the bend angle α . For a bend angle of approximately 263 .
3° and p = p = . α ≤ R min is always smaller than the effective radius R eff for all0 < p ≤ κ R eff profiles for four different values of the bendparameter p for a 90° bend, i.e. p =
0, 0 .
05, 0 .
2, and 1. The path length is normalized to the arclength L = R eff π / p . The first derivative of the curvature is not continuous at thetransition point between the Euler and the circular section. This singularity does not introduceadditional losses because the curvature is continuous and, thus, the mode profile does not changeat this transition. The curvature in the middle of a pure Euler bend ( p =
1) is 82 % higher thanthat of a circular bend with constant radius. This causes increased radiative bend mode lossescompared to a circular bend. On the other hand, for a circular bend the abrupt change in curvatureat the interface between the straight and the bent section induces a mode mismatch. A balancebetween the two extreme geometries optimizes the total loss. (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:4) (cid:2) (cid:1) (cid:5) (cid:2) (cid:1) (cid:6) (cid:2) (cid:1) (cid:7) (cid:3) (cid:1) (cid:2)(cid:2) (cid:1) (cid:2)(cid:2) (cid:1) (cid:4)(cid:2) (cid:1) (cid:5)(cid:2) (cid:1) (cid:6)(cid:2) (cid:1) (cid:7)(cid:3) (cid:1) (cid:2)(cid:3) (cid:1) (cid:4)(cid:3) (cid:1) (cid:5)(cid:3) (cid:1) (cid:6)(cid:3) (cid:1) (cid:7)(cid:4) (cid:1) (cid:2) (cid:3) (cid:8) (cid:9) (cid:7) (cid:2)(cid:1) (cid:5) (cid:11) (cid:9) (cid:12) (cid:4) (cid:10) (cid:11) (cid:9) (cid:6) (cid:1) (cid:2) (cid:1) (cid:1) (cid:2)(cid:2) (cid:8) (cid:16) (cid:18) (cid:14) (cid:2) (cid:1) (cid:17) (cid:9) (cid:19) (cid:12) (cid:1) (cid:13) (cid:10) (cid:15) (cid:11) (cid:19) (cid:12) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:1) (cid:3) (cid:7) (cid:3) (cid:2) (cid:3)(cid:1) (cid:3) (cid:7) (cid:3) (cid:2) (cid:3) (cid:6)(cid:1) (cid:3) (cid:7) (cid:3) (cid:2) (cid:5)(cid:1) (cid:3) (cid:7) (cid:4) (cid:2) (cid:3)
Fig. 2. Normalized curvature along a 90° bend depending on the value of parameter p . Thepath length L is normalized to the arc length L of a circular bend. A pure Euler bend( p =
1) shows an 82 % increased curvature compared to a circular bend ( p = p >
3. Numerical analysis
The modal power and the bend mode radiation in the partial Euler bend geometries were numeri-cally determined with the commercial eigenmode expansion (EME) method tool FIMMPROP(PhotonD, FIMMWAVE) [36]. The software employs a fully vectorial complex finite differencemode solver that allows the calculations of eigenmodes in a bent waveguide section. Theimaginary part of the propagation constant of such an eigenmode reflects its radiative loss. TheEuler section with linearly increasing/decreasing curvature was divided into 64 bend subsectionsof equal path length ∆ s . A single section was used for the circular bend section. The symmetryof the Euler bend at α / .
02 and 1 .
46, respectively [37, 38]. For this generic silicon nitridephotonic platform absorption, scattering, and substrate leakage are neglected because advancedfabrication technologies allow for < − propagation losses in the visible and near-infraredavelength region [8, 27–29]. Therefore, transition and radiative bend mode losses dominatefor bends with small radii. The total path length changes only slightly for different values ofthe bend parameter p with the same effective radius R eff and α < ×
160 nm, which ensures singlemode operation. The influence of the bend geometry on the loss is investigated for the moresensitive TM-like polarized guided mode. Bend loss for TE-like polarization are in generalsmaller compared to the TM-like polarization because the confinement of the TE-like mode ishigher and radiative losses are reduced. Therefore, in PIC designs where both polarizationsare present in the optical path the TM-like polarization typically dictates the minimum usableeffective bend radius. Figures 1(b)-1(d) depict the investigated 45°, 90°, and 180° partial Eulerbend geometries for the bend parameter value p = . p . The results agree within0 .
01 dB, or 7 %, respectively. This confirms that the employed EME method is a suitablealternative to FDTD allowing computationally efficient calculations of waveguide bend losses.
F D T D E M E
Bend loss (dB)
B e n d p a r a m e t e r p S i N 1 6 0 n m7 0 0 n mS i O Fig. 3. Total bend loss determined by eigenmode expansion (EME) and finite-difference-time-domain (FDTD) method for 90° bends, 40 µm effective radius, and TM-like polarization.
Figure 4 summarizes the calculated total bend loss for the guided mode propagation through a45° partial Euler bend with varying bend parameter p . A bend parameter of p = . R eff =
40 µm can bereduced to 0 .
07 dB ( p = .
3) compared to 0 .
09 dB ( p =
0) and 0 .
14 dB ( p = p > . R eff larger than approximately 70 µm. However, for p < . .
001 dB( p = .
2) for an optimized bend to 0 .
026 dB ( p =
0) for a purely circular bend. At R eff =
40 µmthe total bend loss is 0 .
05 dB for p = .
2, compared to 0 .
10 dB for a circular bend ( p = .
19 dB for a pure Euler bend ( p = . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0 00 . 0 50 . 1 00 . 1 5 Bend loss (dB)
B e n d p a r a m e t e r pR e f f ( µ m ) : 4 0 5 0 6 0 7 0 Fig. 4. Total bend loss through a 45° partial Euler bend for different bend radii, TM-likepolarization. Inset depicts the bend geometry.
S-bend structure will be larger than through a single 90° bend for the same effective bend radiusas the number of transitions from straight to bend waveguide sections is doubled in an S-bendstructure. R e f f ( µ m ) : 4 0 5 0 6 0 7 0 Bend loss (dB)
B e n d p a r a m e t e r p ( b )( a ) p = 0 . 0 p = 0 . 2 p = 1 . 0 Bend loss (dB)
E f f e c t i v e r a d i u s R e f f ( µ m ) Fig. 5. Total bend loss through a 90° partial Euler bend, TM-like polarization. (a) Variationof bend parameter p for different effective bend radii. (b) Variation of effective bend radius R eff for different values of the bend parameter p . The area with loss larger than 0 . Figure 5(b) summarizes the total bend loss of a circular bend, an optimized Euler bend( p = . p =
1) for a varying effective radius R eff . The optimized Eulerbend provides the smallest losses over the studied range of effective bend radii. One can observethat for bend radii larger than 50 µm a pure Euler bend is beneficial compared to a circular bend.However, below this radius, the reduced transition losses compared to the circular bend no longercompensate for the increasing radiative bend mode losses from the larger curvature of the pureEuler bend, resulting in a better performance of the circular bend compared to a pure Eulerbend. For a targeted loss value of 0 . p = . p = p = .
026 dB per bend, the partial Euler bend with the same loss has aneffective radius of 43 µm ( p = .
2) and 57 µm ( p = . . R eff =
40 µm and four distinct values of the bend parameter ( p = p = . p = . p = p = .
2) balances radiative bend mode losses arisingfrom sections of high curvature with transition losses from sections of rapidly changing curvature. t o t a l b e n d m o d e t r a n s i t i o n
Loss (dB) p = 0 . 2 ( a ) p = 1 Loss (dB) ( b )
Loss (dB) p = 0 . 0 5 ( c ) Loss (dB)
P a t h l e n g t h ( µ m ) p = 0 ( d ) Fig. 6. Bend mode radiative, transition, and total losses accumulated along 90° partial Eulerbends with effective radius of 40 µm for (a) p =
1, (b) p = .
2, (c) p = .
05, and (d) p = p = .
2) distributes the accumulated loss along the bend. Incontrast, a pure Euler bend ( p =
1) suffers from high radiative losses in the middle of thebend where curvatures are large. The geometry with p = .
05 is close to a circular bend( p =
0) and the transition losses in the Euler sections are the major loss.
Transition losses are not limited to the case of a discontinuous jump in curvature, which occursfor example at the interface between the straight input and bend section of a circular bend, butare present whenever a change in curvature d κ / ds is present. In the circular middle section of partial Euler bend, no transition losses occur ( d κ / ds =
0) and only the radiative bend modecomponents contribute to the loss. In contrast to the optimized bend, the pure Euler bend ( p = p = .
05 is close to a circular bend and has high losses at the start and endof the bend, where the transition loss arising from the fast changing curvature contribute most tothe total loss. On the other hand, radiative bend mode losses are small for this geometry, as theminimal radius of curvature is larger than the minimal radius of curvature of the optimized andpure Euler bend.Like the 90° bend, the 180° bend shows lowest loss for a partial Euler bend with the parameter p around 0 .
2. The bend loss decreases from 0 .
083 dB to 0 .
047 dB for an effective bend radius of R eff =
40 µm. Figure 7 shows the bend loss for effective bend radii ranging from 40 µm to 70 µm.In contrast to 45° and 90° bends, the minimum radius of curvature R min of a 180° bend is closeto a circular bend for all p . This results in a reduced dependence of the bend loss on the partialEuler bend parameter for p > . R e f f ( µ m ) : 4 0 5 0 6 0 7 0 Bend loss (dB)
B e n d p a r a m e t e r p Fig. 7. Total bend loss through 180° Euler bends, TM-like polarization. Inset depicts thebend geometry.
In summary, the findings underline that the implementation of a pure Euler bend and otherclasses of bend geometries with sections of small radius of curvature can introduce pronouncedradiative bend mode losses. Compared to a pure Euler bend a partial Euler bend increases theminimum radius of curvature R min at the expense of transition losses. This trade-off reduces thetotal bend loss. It has to be pointed out that the optimized values for the Euler bend parameter p are specific to the investigated waveguide system. The waveguide geometry, the materialrefractive indices, the wavelength, and the polarization determine the confinement of the guidedmode and, thus, strongly influence the radiation properties of curved waveguides. Therefore, adedicated optimization for a specific waveguide system, effective bend radius, and polarization ismandatory. For a 90° partial Euler bend on silicon photonic platform Fujisawa et al. obtained anoptimal ratio of 68 % between the Euler section length and the total bend length with a radiusof R eff = p = .
52 in our parameterization, which is considerably larger than the value of p = . R eff =
40 µm in our study for a silicon nitride photonic platform. . Experimental analysis
Partial Euler bends of 90° with an effective radius of 50 µm and different values of the bendparameter p have been fabricated on a SiN photonic platform [39]. The SiN waveguide layerhas a nominal thickness of 160 nm. Figure 8(a) shows a microscope image of these structures.The width of the waveguide was measured by scanning electron microscopy to be 680 nm. Thewaveguide cross-section ensures single mode operation. Refractive indices of the SiO claddingand the SiN waveguide material have been determined by ellipsometry to be n SiO2 = .
46 and n SiN = .
92 at a wavelength of 850 nm. The test structures consist of 240 cascaded 90° bends forthe TE-like polarization. Each bend is separated from the subsequent bend by a 10 µm straightsegment. To account for the high bend losses for the TM-like polarization, a cleaved sample withfour bends for the TM-like polarization was prepared.To characterize the bend losses light from a 850 nm Ti:sapphire laser source was end-facecoupled via a polarization maintaining fiber (NA 0 .
12) to the waveguides with the bend structures.The polarization plane of the fiber was adjusted with a mechanical rotator and verified usinga linear polarizer in combination with an optical power meter to couple to the TE-like andTM-like PIC waveguide mode, respectively. The output light of the PIC was collected with asingle mode fiber (NA 0 .
12) connected to an optical power meter. The positioning of the inputand output fibers to the PIC was supported by a piezoelectric auto alignment system (Thorlabs,NanoTrak®). The repeatability of the transmission power level measurement was better than0 . . . Fig. 8. (a) Microscope image of fabricated SiN partial Euler bends with an effective radius50 µm. (b) Near-infrared image of radiation emitted from the bends and scattered at thelithography fill pattern.
The radiation arising from the transition losses and bend waveguide modes for the TM-likepolarization was recorded with an infrared camera (Hamamatsu, IR Vidicon C2741) and isvisualized in Fig. 8(b). The light enters the bend structure from the left handside. Strong radiationpatterns are visible at each bend. Tracing the streaks back to the source suggests the maincontribution from the middle of the bend, where the curvature is largest.igures 9(a) and 9(b) summarize the influence of the bend parameter p on the total bendlosses in TE-like and TM-like polarization, respectively. For the TE-like polarization the modemismatch between the straight and bend section is a major loss component and the largest bendlosses occur, hence, for a circular bend with 0 .
028 dB per bend. With an optimized partial Eulerbend, the bend loss is reduced to 0 .
007 dB at p = .
2. For larger values of the bend parameter p ,the losses start to slowly increase. The numerical simulations with the EME method reflect themeasurements to a high degree. For the TM-like waveguide mode the major losses arise fromthe radiative bend modes. Therefore, the optimal bend geometry converges to a circular bend atwhich the radius of curvature is maximized. A pure Euler bend ( p =
1) is detrimental to the bendperformance. The offset between experiment and numerical analysis for the TM-like polarizationcan most likely be attributed to the effective thin film thickness of the SiN waveguide layer, as thebend losses of the TM-like mode are highly sensitive to variations of the waveguide geometry.For example, a small change of the waveguide height in the simulations from 160 nm to 158 nmincreases the bend loss of the TM-like mode by 0 . (cid:4) (cid:3) (cid:4) (cid:4) (cid:3) (cid:6) (cid:4) (cid:3) (cid:8) (cid:4) (cid:3) (cid:10) (cid:4) (cid:3) (cid:11) (cid:5) (cid:3) (cid:4)(cid:4) (cid:3) (cid:4) (cid:4)(cid:4) (cid:3) (cid:4) (cid:5)(cid:4) (cid:3) (cid:4) (cid:6)(cid:4) (cid:3) (cid:4) (cid:7) (cid:4) (cid:6)(cid:8)(cid:5) (cid:1) (cid:7) (cid:9) (cid:10)(cid:10) (cid:1) (cid:2) (cid:5) (cid:4) (cid:3) (cid:3) (cid:9) (cid:14) (cid:8) (cid:1) (cid:15) (cid:7) (cid:16) (cid:7) (cid:13) (cid:9) (cid:17) (cid:9) (cid:16) (cid:1) (cid:1) (cid:1) (cid:4) (cid:2)(cid:1) (cid:3) (cid:2) (cid:6) (cid:4) (cid:2) (cid:12) (cid:10) (cid:11) (cid:9) (cid:6) (cid:5) (cid:2) (cid:12) (cid:10) (cid:11) (cid:9) (cid:4) (cid:3) (cid:4) (cid:4) (cid:3) (cid:6) (cid:4) (cid:3) (cid:8) (cid:4) (cid:3) (cid:10) (cid:4) (cid:3) (cid:11) (cid:5) (cid:3) (cid:4)(cid:4) (cid:3) (cid:4)(cid:4) (cid:3) (cid:9)(cid:5) (cid:3) (cid:4)(cid:5) (cid:3) (cid:9)(cid:6) (cid:3) (cid:4) (cid:1) (cid:13) (cid:14) (cid:13) (cid:2) (cid:1) (cid:1) (cid:12) (cid:5) (cid:9) (cid:11) (cid:1) (cid:18) (cid:17)(cid:1) (cid:13) (cid:14) (cid:13) (cid:2) (cid:1) (cid:1) (cid:12) (cid:5) (cid:10) (cid:4) (cid:1) (cid:18) (cid:17)(cid:1) (cid:17) (cid:16) (cid:15) (cid:20) (cid:22) (cid:19) (cid:16) (cid:17) (cid:16) (cid:18) (cid:21) (cid:4) (cid:6)(cid:8)(cid:5) (cid:1) (cid:7) (cid:9) (cid:10)(cid:10) (cid:1) (cid:2) (cid:5) (cid:4) (cid:3) (cid:3) (cid:9) (cid:14) (cid:8) (cid:1) (cid:15) (cid:7) (cid:16) (cid:7) (cid:13) (cid:9) (cid:17) (cid:9) (cid:16) (cid:1) (cid:1) Fig. 9. Total bend loss measurement and numerical results through 90° partial Euler bendswith effective radius of 50 µm for (a) TE-like and (b) TM-like waveguide mode. Insets showthe bend waveguide profile for the major electric field components at the smallest radius ofcurvature of an Euler bend.
The SiN of the employed PIC technology platform has a refractive index which is 5 % smallerthan the value taken from literature and used in the numerical analysis in the previous section.This moderate difference in refractive index causes a strong increase of the minimum bendloss for the 50 µm bends in the TM-like polarization from 0 .
01 dB to 0 .
5. Conclusion
In this work the major loss mechanisms for partial Euler bend geometries were numericallyanalyzed with the EME method. This method provides additional insights, not accessible withther tools, into the individual contributions to the total loss along the geometry. Optimal valuesfor the bend parameter have been identified for 45°, 90° and 180° silicon nitride waveguidepartial Euler bends operated with TM-polarization using literature values for the refractiveindices at an operation wavelength of 850 nm. The proper choice of the bend parameter balancestransition losses arising from the continuous transition between sections of changing curvature,and radiative losses originating from the curvature of the bend waveguide mode. Transition lossesare not limited to the abrupt change between a straight and a bent section, but occur wheneverthe curvature is changing non-adiabatically, e.g. for all bends with a significantly fast change incurvature. Compared to a pure Euler bend without a section of constant curvature, an optimizedpartial Euler bend exhibits a reduced bend loss. With a partial Euler waveguide bend geometrythe integration density of PICs can be increased and losses for complex designs relying on a largenumber of bends reduced. The numerical analysis has been applied to fabricated partial Eulerbends for a specific silicon nitride photonic platform. The measurements of the bend losses agreewell with the EME simulations and confirm the feasibility of this method to optimize waveguidebends for a specific material system and waveguide geometry. The small differences betweenthe refractive indices assumed in the theoretical study and the experimental photonic platformshowed a considerable impact on the optimal value of the bend parameter and on the actual lossvalues. Our findings emphasize that the optimal ratio between the Euler section to the total bendlength has to be determined individually for different waveguide systems.
Funding
Austrian Research Promotion Agency (FFG) (850649); European Union’s Horizon 2020 researchand innovation programme (688173).
Acknowledgments
This research has received funding through the grant PASSION (No. 850649) from the AustrianResearch Promotion Agency (FFG) and European Union’s Horizon 2020 research and innovationprogramme under the grant OCTCHIP (No. 688173).
Disclosures
The authors declare no conflicts of interest.
References
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