Design of a Broadband Integrated Mode Converter Based on Non-Uniform Binary Long-Period Grating by Direct Multiparameter Optimization

We present a novel approach for designing non-uniform binary long-period gratings with predefined transmission characteristics by directly optimizing the coupling strengths and lengths of the constituent waveguides in all grating segments. The optimal grating parameters were determined by minimizing the deviation of the transmission characteristics calculated using the transfer matrix method from the predefined target functions using the multistart optimization procedure. We demonstrate the effectiveness of the proposed method by optimizing the performance of grating-based TE0/TE1 mode converters in a few-mode buried waveguide composed of medium-contrast materials (TiO2:SiO2/SiO2). Two converters with coupling caused by the lateral shift and step-like thickness change of adjacent waveguides were numerically optimized for the 1.3 and 1.55 μm bands. Each of the optimized devices has a total length of approximately 250 μm. For both types of converters, a spectral range of 150 nm with crosstalk lower than −20 dB was obtained for the two communication windows using at most nine grating segments, with twice lower excess loss for the grating with a step-like thickness change. Moreover, for such grating composed of only 9 segments it is possible to obtain the conversion band of 400 nm, covering both communication windows with an excess loss lower than 0.45 dB and crosstalk below −20 dB. We also numerically analyzed the impact of the fabrication tolerances on the converter characteristics.


Design of a Broadband Integrated Mode Converter Based on Non-Uniform Binary Long-Period Grating by Direct Multiparameter Optimization
Edyta Środa , Jacek Olszewski , and Waclaw Urbanczyk Abstract-We present a novel approach for designing nonuniform binary long-period gratings with predefined transmission characteristics by directly optimizing the coupling strengths and lengths of the constituent waveguides in all grating segments.The optimal grating parameters were determined by minimizing the deviation of the transmission characteristics calculated using the transfer matrix method from the predefined target functions using the multistart optimization procedure.We demonstrate the effectiveness of the proposed method by optimizing the performance of grating-based TE 0 /TE 1 mode converters in a fewmode buried waveguide composed of medium-contrast materials (TiO 2 :SiO 2 /SiO 2 ).Two converters with coupling caused by the lateral shift and step-like thickness change of adjacent waveguides were numerically optimized for the 1.3 and 1.55 µm bands.Each of the optimized devices has a total length of approximately 250 µm.For both types of converters, a spectral range of 150 nm with crosstalk lower than −20 dB was obtained for the two communication windows using at most nine grating segments, with twice lower excess loss for the grating with a step-like thickness change.Moreover, for such grating composed of only 9 segments it is possible to obtain the conversion band of 400 nm, covering both communication windows with an excess loss lower than 0.45 dB and crosstalk below −20 dB.We also numerically analyzed the impact of the fabrication tolerances on the converter characteristics.
Digital Object Identifier 10.1109/JPHOT.2024.3373751copropagating core modes.Due to the resonant nature of the couplings, LPGs typically have a narrow spectral bandwidth, which is an important limitation in many practical applications.
In the case of channel waveguides, the TE 0 /TE 1 conversion bandwidths that have been reported are equal to 4 nm (−20 dB CT) for the top surface corrugated polymer waveguide LPG [10], 20 nm (−20 dB CT) for sidewall corrugated polymer waveguide LPG [11], approximately 80 nm (−20 dB CT) for polymer waveguide cascaded LPG [12], 150 nm and 300 nm (−20 dB CT) to two TE 1 modes with different symmetry for phase-shifted length-apodized polymer waveguide LPGs [9], approximately 70 nm (−20 dB CT) to the first order orthogonally polarized modes for two-period embedded polymer waveguide LPG [13], and approximately 50 nm (−20 dB CT) to the first order mode for the polymer waveguide LPG with corrugated top surface [14].
The second significant limitation of LPGs is their large span, caused by the length of the period required to obtain phase matching, which in medium-and low-contrast waveguides ranges from several tens to several hundred micrometers, yielding an LPG length from a single millimeter to tens of centimeters.Therefore, achieving the desired transmission characteristics with a minimal number of grating segments is essential for many practical applications.However, this aspect has rarely been considered.For example, a conversion bandwidth of 145 nm for the LP 01 and LP 11 modes was obtained in a fiber LPG with four phase-shifted chirped segments composed of 40 periods and a total length of approximately 4.62 mm [4].In [9], a bandwidth of 150 nm (−20 dB CT) was obtained in a length-apodized LPG fabricated in a polymer waveguide grating composed of 24 periods with a total length of 7.44 mm.In [13], a two-period embedded polymer waveguide LPG of ultra-short length of 824 μm for broadband mode conversion (75 nm bandwidth with −20 dB CT) was demonstrated.
The methods reported in the literature for designing LPGs with predefined transmission characteristics are based on the inverse scattering approach [21], [22] or direct optimization [23], [24], [25], [26], [27], [28], [29].In the first method, the grating profile is directly calculated from the desired transmission spectrum.Whereas, in the latter method, the difference between the calculated and targeted characteristics is minimized by modifying the set of fiber/waveguide geometrical parameters using different optimization approaches, including genetic algorithms [23], evolutionary programming [24], [25], variational-based Lagrange multiplier optimization [26], [27], and particle swarm optimization [28], [29].Note that all these methods are based on the coupled-mode theory, assuming weak coupling.The advantage of weakly coupled LPGs is their low excess loss; however, at the expense of large length, as tens of grating periods are typically required to obtain the target characteristic.
The number of grating segments required to achieve full power transfer between converted modes for a uniform grating is inversely proportional to the coupling strength.Therefore, one may expect that by increasing the coupling strength and choosing the appropriate segments lengths, the required transmission characteristics can be obtained with a significantly reduced number of grating periods.We adopted the transfer matrix approach based on the mode expansion method proposed in [30] to model strongly coupled binary non-uniform gratings, even in the case of few-mode propagation.Using this formalism, we analyzed LPGs made on buried-channel waveguides, for which the length of each waveguide and the coupling strength at each interface between successive waveguides can be controlled during the fabrication process by varying their geometry or transverse shift.The transmission characteristics of such gratings for each input spatial mode are expressed as functions of their structural parameters (lengths of all segments and coupling strengths at all interfaces), and the search for the best grating structure is reduced to solving a purely numerical optimization problem using the parallel multi-start method.As a merit function, we used the square root deviation of the calculated transmission characteristics relative to the predefined target characteristics for a particular input mode.Optimization was performed for a subsequent increase in the number of grating segments until the target characteristics were achieved with satisfactory accuracy.The grating obtained in this manner meets the target characteristics using as few structural elements as possible.
Notably, in integrated optics technology, devices other than LPGs are also used for mode conversion -a comprehensive review of current solutions in this field in terms of bandwidth, crosstalk, and devices' footprint can be found in [31].In particular, two multimode converters based on subwavelength structures optimized numerically using a genetic algorithm [32] and the particle swarm method [33] were recently reported for silicon platforms.Such devices have a considerably small footprint (respectively, 1.8 μm × 2.2 μm and 1.3 μm × 2.7 μm) and wide spectral conversion bands (335 nm and 407 nm, respectively, with -10 dB CT), but they require advanced fabrication technology; in the case of [32], they have a significant excess loss.
The approach we propose, consisting of the direct optimization of a non-uniform binary grating structure, allows achieving a record-breaking TE 0 /TE 1 conversion efficiency in the class of LPG-based devices, which are composed of a small number of segments.To demonstrate the effectiveness of the proposed approach, two types of grating converters with coupling caused by a lateral shift and a step-like thickness change of adjacent waveguides were numerically optimized for 1.3 and 1.55 μm bands.For both types of gratings, a spectral conversion range of 150 nm with CT significantly lower than −20 dB can be obtained for the two communication windows using at most 9 grating segments, with twice lower excess loss for the structure with a step-like thickness change (below 0.19 dB and 0.37 dB, respectively in the 1.3 μm and 1.55 μm bands).Moreover, for the grating of this type composed of only 9 segments, it is possible to obtain the conversion band of 400 nm, covering both communication windows with an excess loss lower than 0.45 dB in the full spectral range and CT below −20 dB.Furthermore, we confirmed numerically the acceptable fabrication tolerances for both grating types.

II. TRANSFER MATRIX METHOD FOR NON-UNIFORM BINARY GRATING
To calculate the transmission characteristics of a few-mode non-uniform binary waveguide grating, we used the transfer matrix method based on a mode-matching technique [30].The structure of the grating, consisting of N segments, is shown in Fig. 1.All segments, except the last, are composed of waveguides of types a and b, wherein the a-type waveguide has the same geometry in all grating segments, whereas the b-type waveguide is a perturbed form of the a-type waveguide.The last grating segment is composed of only waveguide b because the leading-in and leading-out waveguides are of type a, as shown in Fig. 1.The perturbation of waveguide b may assume the form of a transverse shift or a slight geometry modification that can change from segment to segment; thus, allowing variation in the coupling coefficients between the propagating core modes.
Assuming that both waveguides support M spatial modes and neglecting back reflections and couplings to cladding modes, the coupling between j-th core mode propagating in waveguide a and i-th core mode propagating in waveguide b at the k-th b/a interface can be expressed by the following M × M matrix of coupling coefficients: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where (Ea ) represent the electric and magnetic field distributions, respectively, in the j-th mode of waveguide a and i-th mode of waveguide b, both carrying power normalized to unity (i, j = 1 corresponds to the fundamental core mode TE 0 ), and z is the unit vector in the direction of propagation.Correspondingly, the coupling matrix Cab at the interface a/b is expressed by the Hermitian conjugate of the Cba matrix.
The field distributions at the grating input/output can be represented as the sum of the normalized waveguide eigenmodes, with weighting coefficients expressed by the column vectors where |a j | 2 represents the relative power carried by the j-th eigenmode.For a grating composed of N waveguides of type b and N-1 of type a (Fig. 1), the two vectors representing the input/output fields are related by the following matrix formula: (2) where P a (k) and P b (k) represent the propagation matrices for waveguides of types a and b in the k-th grating segment, with non-zero elements only on the diagonal: where βa are the real and imaginary parts of the complex propagation constants β = β − iα for the waveguides of type a and b in the k-th grating segment, whereasLa (k) , Lb (k) are the lengths of respective waveguides in the k-th segment.The confinement loss γ of a particular mode is related to the imaginary part of its complex propagation constant α as follows: γ = 20α/ln (10).
This representation of the binary grating considers the loss for each of the modes and enables optimization of the multimode grating structure, including the spatial modes near the cutoff.Moreover, the grating performance for each of the excited modes can be described using the transmission (T ii ) and cross-coupling (T ji ) coefficients to the other modes, defined as follows: The grating transmission characteristics defined in this manner depend on the lengths of waveguides a and b in every segment (La (1) , . . ., La (N −1) , Lb (1) , . . ., Lb (N ) ) and the coupling matrices Cab (1) (s (1) ), . . ., Cab (N ) (s (N ) ) being a function of the perturbation strength s (k) of waveguide b with respect to a.

III. FORMULATION OF THE OPTIMIZATION PROBLEM
In our approach, designing a binary grating with the required transmission characteristics for the respective modes is a purely numerical optimization problem.It consists of finding the grating parameters that minimize the merit function, which can be expressed as the square root deviation of the calculated characteristics with respect to the specific target characteristics predefined for the input mode of the i-th number: (1) , . . ., La (N −1) , Lb (1) , . . ., Lb (N ) , s (1) , . . ., s (N ) where q ji are the weighting coefficients determining the preferences for specific grating characteristics, s (k) represent the perturbation strength at the respective segments, and m j determines the density of the spectral sampling.For example, to obtain the grating parameters fabricated in the waveguide supporting the three spatial modes (TE 0 , TE 1 , and TE 2 ) that ensure the effective conversion of the TE 0 (i = 1) input mode to the TE 1 mode (j = 2) and simultaneously minimize the cross-coupling to TE 2 mode (j = 3), the relevant parameters in the above expression over the spectral range of interest should be set as follows: Such a merit function structure minimizes the power at the grating output in the TE 0 and TE 2 modes while maintaining the power in the targeted mode (TE 1 ) at a possibly high level.Setting the weighting factor q 21 = 0.1 causes all the terms of the merit function to contribute to its minimized value in a balanced manner.
In general, all grating parameters can be optimized.From a technological perspective, it is relatively easy to change the waveguide lengths La (k) and Lb (k) for the individual grating segments.Tuning the coupling strengths s (k) requires individually adjusting the perturbation magnitude in each segment, which depends on the specific grating structure and can be realized by varying the lateral position, height, width, or crosssectional shape of the waveguide of type b relative to the unperturbed waveguide a.
Our optimization procedure utilizes primarily a built-in Mul-tiStart Algorithm available in MATLAB's Global Optimization Toolbox.In the first step, the initial number of grating segments N, specific target characteristics T target ij (λ) and merit function are defined.Subsequently, the algorithm randomly generates a set of starting points by considering the defined bounds for each structural parameter of the grating.In the second step, local solvers are run to search for the local minima of the merit function expressed by ( 6) (exploiting the built-in parallel processing feature, as the local minimum for each generated starting point is searched independently) until the stopping conditions are satisfied, after which the best candidate for the global minimum from the determined set of local minima is returned.The procedure for searching for a global minimum is restarted in a loop until the returned candidate for the global minimum stabilizes, yielding the minimum deviation from the target characteristics.The solution found in this manner is a vector composed of independent grating parameters, that is, the lengths of all segments (La (k) and Lb (k) ) and the perturbation strengths s (k) determining the elements of the coupling matrices for all interfaces.If, for the initial number of grating segments N, the set of optimal parameters does not ensure the achievement of the target characteristics with an acceptable square root deviation, the optimization procedure is successively repeated for gradually increased N until the target characteristics are reached with satisfactory precision.Note that the grating structure determined in this manner is optimal because the target characteristics are achieved with the desired accuracy using the smallest possible number of grating segments.

IV. OPTIMIZED BROAD-BAND TE
The effectiveness of the proposed approach was confirmed by optimizing the design of two non-uniform binary long-period gratings made of a medium-contrast buried channel waveguide (TiO 2 :SiO 2 /SiO 2 ) to ensure broadband conversion between the TE 0 (i = 1) and TE 1 (j = 2) modes.
In both cases, the optimization procedure considered the couplings between several spatial modes guided by the grating.The quality of the optimized device was quantified using three parameters, conversion efficiency (CE), crosstalk (CT), and excess loss (EL), defined for a few-mode TE 0 /TE 1 grating-based converter as follows:

A. Grating With Coupling Induced by a Lateral Offset Between Adjacent Waveguides
The detailed structure of the first analyzed grating, composed of N segments, is shown in Fig. 1(a).Except the last one, all segments consist of two waveguides of the same dimensions (W = 2.5 μm, d = 0.42 μm) and different lengths La (k) and Lb (k) , with the b-type waveguide transversely shifted by Δx (k) relative to the waveguide a.We assume that the lateral offset Δx (k) may be different in each segment and its maximum value is arbitrarily set at Δx max = 0.22 μm to limit the excess loss of the device to a level not exceeding 1 dB.
In Fig. 2(a), (b), we show the spectral dependence of effective indices and confinement loss calculated using finite element method (FEM) and scattering boundary conditions (SBCs) for the 6 lowest order spatial modes in the spectral range from 1.2 to 2.0 μm.These calculations consider the spectral dependence of refractive indices of both materials (TiO 2 :SiO 2 in the core and SiO 2 in the cladding) based on experimental data available in the literature [34], [35].The refractive indices for such glasses are equal n TiO2:SiO2 = 1.7629, n SiO2 = 1.4469 for λ = 1.30μm, and n TiO2:SiO2 = 1.7565, n SiO2 = 1.4440 for λ = 1.55 μm.
SBCs are built-in boundary conditions approximating the Sommerfeld radiation condition in the COMSOL Multiphysics FEM software.We chose SBCs rather than perfectly matched layers because confinement loss calculated with both approaches differ by less than an order of magnitude while SBCs perform much faster.
The chosen waveguide dimensions support three TE modes (TE 0 , TE 1, and TE 2 ) and three TM modes (TM 0 , TM 1, TM 2 ) in the analyzed spectral range.We show in Fig. S1 in the Supplementary material the calculated self and cross-coupling coefficients for all combinations of TE modes versus wavelength Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.for Δx max = 0.22 μm and versus Δx for λ = 1.3 μm.Because waveguides a and b both have two-plane symmetry, the cross-coupling coefficients for any combination of TE and TM modes are negligibly small (below 10 -6 ).Therefore, we limit the propagation analysis to TE-type modes only, which reduces the dimensions of the coupling matrices Cba (k) to 3×3.The grating optimization procedure was conducted for the spectral range (λ 0 -Δλ, λ 0 +Δλ) with λ 0 set at 1.3 and 1.55 μm and the bandwidth 2Δλ = 0.15 μm.The loss increase of the TE 2 mode near the cut-off wavelength (λ c = 1.47 μm) was considered during the optimization by substituting to the propagation matrix expressed by (4) the imaginary part of the propagation constant α calculated using the SBCs (Fig. 2(b)), whereas the other spatial modes were considered lossless (α = 0).To maximize the conversion of the initially excited TE 0 mode to the TE 1 mode, and simultaneously maintain a low crosscoupling to the TE 2 mode, we used the merit function expressed by (6), with a spectral sampling step of 10 nm.To maintain the converter reasonably short, the possibility of varying the lengths of the respective waveguides during the optimization procedure was limited to the range from 0.2L π a/b to 2L π a/b , where L π a/b is the waveguide length that ensures resonant coupling for the central wavelength λ 0 .As a starting point for the optimization procedure, we used the randomly generated set of parameters: Δx (k) , La (k) and Lb (k) for k = 1 to N, by considering the defined bounds for each structural parameter of the grating.The grating structure was optimized for a gradually increasing number of segments N until the required characteristics were obtained, that is, a CT below −20 dB and an EL lower than 1 dB in the target band of the device.The results of the optimization of the TE 0 /TE 1 mode converter for the wavelength range of 1.3 ± 0.075 μm with the weight factors in the merit function set at q 11 = q 31 = 1, q 21 = 0.1 are shown in Fig. 3.For this spectral window, the required characteristics were achieved for a grating nominally composed of N = 8 segments, which corresponds to a total grating length L tot = 254.5 μm.As shown in Fig. 3(a), the fraction of power remaining in the initially excited TE 0 mode, as well as the residual power coupled to TE 2 mode are considerably below −20 dB within the target band.Moreover, the conversion efficiency to the TE 1 mode is greater than 89% in the entire band and equals to 92% In Fig. 4, we show the optimization results for the TE 1 /TE 0 converter for the range 1.55 ± 0.075 μm with the nominal number of segments set at N = 9.The converter is effectively twomode in this spectral window because the TE 2 mode is highly attenuated.In this case, because of the decrease in the coupling coefficient between the TE 0 /TE 1 modes at longer wavelengths (Supplementary material, Fig. S1 a), the optimization procedure does not disable any of the segments, and the effective number of segments is equal to nine, as shown in Fig. 4(b).As it is shown in Fig. 4(a), the greater number of grating segments and extinction of the mode TE 2 (not guided in this spectral range) lowers the TE 0 /TE 1 conversion efficiency to approximately 85% at the central wavelength (excess loss 0.72 dB), whereas the crosstalk remains considerably below −20 dB over the entire spectral range.In this case, the optimal lengths of waveguides a and b deviate only slightly with respect to L π a/b = 13.164μm.This is caused by an almost linear dependence of the difference between the effective refractive indices for the TE 0 and TE 1 modes in the 1.55 ± 0.075 μm spectral range, which yields a wide conversion band for nearly constant lengths of all segments.However, for the optimal grating structure composed of such waveguides, the lateral offset Δx (k) gradually decreases with the segment number, which minimizes the excess losses while maintaining a high conversion purity with a CT considerably below −20 dB in the target band.

B. Grating With Coupling Induced by Step-Like Thickness Decrease
As a second example, we optimized the grating with the perturbation of waveguide b in the form of a step-like thickness decrease over part of its width, as shown in Fig. 1(b).Similar to the previous case, waveguides a and b are made of medium-contrast materials (TiO 2 :SiO 2 /SiO 2 ) but have different cross-sections.The unperturbed waveguide a is of rectangular shape of the same dimensions as in the previous case (2.5×0.42 μm), whereas the width of the step-like thickness decrease in the perturbed waveguide b is tuned within the range 0 ≤ w ≤ W/2 to control the coupling strength, where W = 2.5 μm is the waveguide width.The step depth is constant in all segments (t = 70 nm) and selected in such a way that the maximal coupling coefficient between TE 0 and TE 1 modes arising for the step width w = W/2 is similar to the coupling in the grating with lateral offset for Δx max = 0.22 μm (∼0.2 at λ = 1.3 μm).
In Fig. 5, we show the spectral dependence of effective indices and confinement loss calculated for the unperturbed (type a) and perturbed (type b) waveguides for all guided spatial modes (TE 0 , TE 1 , TE 2, and TM 0 , TM 1 , TM 2 ) in the spectral range from 1.2 to 2.0 μm.In contrast to the grating with a lateral offset, the perturbation in the form of a step-like thickness decrease cancels the two-plane symmetry of waveguide b and causes the coupling coefficients (see Supplementary material, Fig. S2) between the TE and TM modes at each a/b interface to become nonzero, potentially resulting in a noticeable power transfer from the excited TE 0 mode to the TM modes.Therefore, the TM modes must be included in the propagation analysis, which increases the dimensionality of the optimization procedure to six spatial modes.
The limits imposed on the variation of lengths La (k) , Lb (k) are the same as those for the optimization of the grating with a lateral offset, while the weight coefficients in the merit function are set as follows: q 11 = q 21 = q 41 = q 51 = q 61 = 1 and q 31 = 0.1.As a starting point for the optimization procedure, we used the randomly generated set of parameters: w (k) , La (k) and Lb (k) for k = 1 to N by considering the defined bounds for each structural parameter of the grating.
The results of the optimization of the TE 0 /TE 1 mode converter for the wavelength range of 1.3 ± 0.075 μm are shown in Fig. 6.Similar to the grating with an offset, the optimization was started for a nominal number of segments N = 8; however, crosstalk significantly lower than −20 dB was reached for the grating composed effectively of only N = 6 segments because the optimization process disabled segments 2 and 3 (w = 0 for these two segments).Fig. 6 shows that CT is limited mostly by the power remaining in the initially excited TE 0 mode.Whereas, the power converted to other spatial modes, including both TM modes, is considerably lower in most of the target spectral range.The conversion efficiency to the TE 1 mode and the excess loss are better than for the converter with lateral offset and equal respectively to 95% and 0.19 dB at the central wavelength of 1.3 μm, compared to 92% and 0.36 dB for the grating with offset.
For the spectral range 1.55 ± 0.075 μm, the optimization procedure does not disable any of the segments, and the effective number of segments remains equal to 9 (Fig. 7(b)), yielding the following converter parameters: CT below −20 dB in the target wavelength range, CE greater than 90% in the entire spectral window (91,8% at the central wavelength of 1.55 μm, EL of 0.37 dB).These characteristics are better than those of the converter with lateral offset and the same number of segments (N = 9), for which the CE and EL at 1.55 μm are respectively equal to 85% and 0.72 dB, as shown in Fig. 4(a).The systematically lower excess loss of the converter with a step-like thickness decrease is owing to the lower loss at a single interface for this type of perturbation, which for the TE 0 input mode can be expressed as follows for the k-th interface: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.For the perturbation strengths yielding similar coupling between the TE 0 and TE 1 modes equal to 0.2 at 1.3 μm (for Δx = 0.22 μm and w = W/2), the excess loss at single interface calculated according to the above formula equals respectively to 0.020 dB and 0.015 dB for the grating with offset and step-like thickness decrease, which results in better performance of the later one.
Finally, we demonstrated the possibility of designing a TE 0 /TE 1 converter with an ultrawide band of 400 nm, extending from 1225 nm to 1625 nm, which covers both telecommunication windows, as shown in Fig. 8.The found grating structure consists of 9 segments and ensures CT below −20 dB and CE to TE 1 mode better than 90% (EL below 0.45 dB) over the entire spectral range.It is worth emphasizing that such an optimized non-uniform grating has an almost five times wider conversion band compared to the reference uniform grating shown Fig. S3b in the Supplementary material.

C. Analysis of Fabrication Tolerances
We numerically analyzed the effects of manufacturing errors on the characteristics of both types of converters.The converter tolerance to deviations in the segments' lengths was assessed in a manner that random values uniformly distributed in the range from −0.2 to +0.2 μm were added to the lengths La (k) and Lb (k) found in the optimization procedure, and the transmission characteristics for such disturbed structures were calculated 100 times while maintaining other grating parameters unchanged.The results of simulations conducted for both converters designed for the range 1.3 ± 0.075 μm are displayed in Fig. 9(a), (b).They clearly show that random changes in the segment lengths in the range of ±0.2 μm do not significantly deteriorate the performance of either converter.The crosstalk stays below −20 dB throughout the target spectral range and only slightly increases above this limit at the edges.
Moreover, in Fig. 9(c) we show the characteristics of the converter with the step-like thickness decrease calculated for the step depth t changed by ±5%, while in Fig. 9(d), (e) for the converter with lateral offset with the offset Δx and thickness d of waveguides a and b changed by ±5% with respect to the optimal values.These results also prove that both converters are relatively tolerant to fabrication inaccuracies of such a magnitude, that is, the CT remains close to the required −20 dB in the entire spectral range, while the EL is increased maximally to 0.7 dB at its edges.
The simulation results shown Fig. 9(f) indicate that the most critical technological parameter is the waveguide width W because its change by ±5% leads to a significant increase in excess loss (above 1 dB at the edges of the target conversion band).The crosstalk is also increased, but it does not exceed the acceptable value of −20dB.This leads to the conclusion that the waveguide width W should be controlled in the fabrication process with a precision of ±2%.

V. CONCLUSION
We showed that binary nonuniform long-period gratings with strong coupling between different modes at successive interfaces can be used as broadband mode converters.In addition, we proposed a method for directly optimizing such structures to obtain the required conversion characteristics using as few grating segments as possible.To enable the modeling of strongly coupled binary gratings, even in the case of multimode propagation, we adapted the mode expansion method and expressed the power transferred to the respective modes at the grating output as a function of its structural parameters, that is, the lengths of all the segments and the perturbation strengths.In our approach, designing the converter becomes a purely numerical optimization problem that consists of finding a set of structural parameters that minimizes the deviation of the calculated transmission characteristics with respect to predefined target functions.
The effectiveness of the proposed method was proven by optimizing two TE 1 /TE 0 broadband mode converters dedicated for the 1.3 μm and 1.55 μm bands, made of medium contrast (TiO 2 :SiO 2 /SiO 2 ) few-mode waveguides with strong coupling induced by lateral offset or step-like thickness decrease.For both types of converters, an operation band of 150 nm with crosstalk considerably lower than −20 dB was obtained for both spectral ranges using a small number of segments, that is, from six to nine, depending on the spectral window and perturbation type.The converter with step-like perturbation of waveguide thickness shows systematically smaller excess losses in both spectral bands, which are equal to 0.19 dB at 1.3 μm and 0.37 dB at 1.55 μm against 0.36 dB and 0.72 dB for the converter with offset segments.Moreover, for the converter with thickness perturbation composed of only nine segments, it is possible to obtain a very broad conversion band (400 nm) spreading over both communication windows with an excess loss below 0.45 dB and crosstalk below −20 dB.The better performance of the converter with thickness perturbation was caused by a lower excess loss at the single interface compared with the interface between the offset waveguides ensuring the same coupling coefficient between the TE 1 /TE 0 modes.It is worth emphasizing that each of the optimized devices has a total length of approximately 250 μm, which is more than an order of magnitude shorter compared to other solutions known from the literature based on weak couplings.
Furthermore, we performed the tolerances analysis showing that random errors in segments lengths in the range of ±0.2 μm and systematic fabrication errors in waveguide dimensions up to ±5% (except the waveguide width, which should be controlled with an accuracy of ±2%) do not significantly worsen the characteristics of both converters, i.e., CT remains below −20 dB in almost the entire target band, and only a slight increase in EL is observed at its edges.Therefore, because of the reduced number of segments and a significantly broad operation range, the proposed converter structure based on non-uniform, strongly coupled binary LPGs, may be an alternative to the solutions reported in the literature based on weakly coupled gratings, which require a greater number of segments, and consequently are of greater length, especially in the case of medium-and low-refractive-index contrast material platforms.

Fig. 1 .
Fig. 1.Structure of the considered non-uniform binary long-period grating with perturbation between successive waveguides in the form of lateral shift (a) and step-like height decrease (b).

Fig. 2 .
Fig. 2. Propagation characteristics of the buried channel waveguide of dimensions (2.5×0.42 μm) made of TiO 2 :SiO 2 /SiO 2 used for designing the TE 0 /TE 1 mode converter with lateral offset: spectral dependence of effective indices for all spatial modes (a), confinement loss γ and imaginary part of the propagation constants α (b) for the TE modes only.

Fig. 3 .
Fig. 3. Coupling coefficients to respective modes and conversion efficiency to TE 1 mode calculated for the TE 1 /TE 0 mode converter optimized for the 1.3 ± 0.075 μm window (gray area) (a) and layout of the optimal grating structure composed nominally of N = 8 segments, L tot = 254.5 μm (b).Optimal values of the lateral shift between waveguides of type a and b in successive segments (c), optimal lengths of waveguides a and b normalized to L π a/b (d) and (e).

Fig. 4 .
Fig. 4. Coupling coefficients to respective modes and conversion efficiency to TE 1 mode calculated for the TE 1 /TE 0 mode converter optimized for the 1.55 ± 0.075 μm window (gray area) (a) and layout of the optimal grating structure composed nominally of N = 9 segments, L tot = 223.9μm (b).Optimal values of the lateral shift between waveguides of type a and b in successive segments (c), optimal lengths of waveguides a and b normalized to L π a/b (d) and (e).

Fig. 5 .
Fig. 5. Propagation characteristics of medium contrast buried waveguides (TiO 2 :SiO 2 /SiO 2 ) of type a with rectangular cross-section of dimensions W = 2.5 μm, d = 0.42 μm (solid lines) and perturbed waveguide of type b with step-like thickness decrease of dimensions w = W/2, t = 70 nm (dashed lines) used for designing the TE 0 /TE 1 mode converter: spectral dependence of effective indices (a), confinement loss and imaginary part of the propagation constants (b) calculated for all spatial modes.

Fig. 6 .
Fig. 6.Coupling coefficients to respective modes and conversion efficiency to TE 1 mode calculated for the TE 1 /TE 0 mode converter for the 1.3 ± 0.075 μm window (gray area) (a) and layout of the optimal grating structure composed nominally of N = 8 segments, L tot = 222.1 μm (b).Optimal widths of the step-like thickness decrease for the perturbed waveguide b normalized to its width W (c), optimal lengths of a and b waveguides normalized to L π a and L π b

Fig. 7 .
Fig. 7. Coupling coefficients to respective modes and conversion efficiency to TE 1 mode calculated for the TE 1 /TE 0 mode converter for the 1.55 ± 0.075 μm window (gray area) (a) and layout of the optimal grating structure composed nominally of N = 9 segments, L tot = 223.8μm (b).Optimal widths of the step-like thickness decrease for the perturbed waveguide b normalized to its width W (c), optimal lengths of the waveguides a and b normalized respectively to L π a and L π b ensuring resonant coupling for the central wavelength λ 0 (d) and (e).

Fig. 8 .
Fig. 8. Spectral characteristics of the TE 1 /TE 0 ultra-broadband grating converter for the 1.225−1.625μm window (gray area) (a) and layout of the optimal grating structure composed nominally of N = 9 segments, L tot = 221.1 μm (b).Optimal widths of the step-like thickness decrease for the perturbed waveguide b normalized to its width W (c), optimal lengths of the waveguides a and b normalized respectively to L π a and L π b (d) and (e).

Fig. 9 .
Fig. 9. Coupling coefficients to respective modes and conversion efficiency to TE 1 mode calculated for the mode converters with the step-like thickness decrease (a) and segments offset (b) optimized for the range 1.3 ± 0.075 μm (gray area) with perturbed segment lengths by uniformly distributed random values from the range of ±0.2 μm.Characteristics of the converter with the step-like thickness decrease calculated for the step depth t changed by ±5% (c) and the converter with lateral offset with the offset Δx, thickness d, and width W of waveguides a and b changed by ±5% with respect to the optimal values (d)-(f).Other structural parameters of the grating remained unchanged in respective simulations.