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Short-length and broadband mismatched optical couplers with tapered Hamming function for C- and L-band
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Optik
Optics
Optik 121 (2010) 831–838 www.elsevier.de/ijleo
Short-length and broadband mismatched optical couplers with tapered Hamming function for C- and L-band Yun-Sheng Kua, Chi-Feng Chena,b,, Chang-Neng Shauoc, Hsiou-Jeng Shyc a
Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC Institute of Opto-Mechatronics Engineering, National Central University, Jhongli 32054, Taiwan, ROC c Materials & Electro-Optics Research Division, Chung Shan Institute of Science and Technology, P.O. Box 90008-8-8, Lung-Tan, Tao-Yuan, 325 Taiwan, ROC b
Received 5 May 2008; accepted 28 September 2008
Abstract Mismatched optical couplers with variable widths of waveguide tapered by Hamming function are numerically investigated in the demand of short-length, broadband, and low crosstalk. We used global search algorithm and beam propagation method to seek optimal structure parameters of coupling waveguide. The coupler length is 3.6 mm within the C+L-band (1.53–1.61 mm) for variable widths of waveguide at crosstalk level of 35 dB. Comparison with constant width of waveguide, the constant width of waveguide has a coupler length of 4.4 mm and can only achieve 20 dB of crosstalk within the C-band (1.53–1.565 mm). Obviously, the waveguide with variable widths has the advantage over constant width for the demand of short-length, bandwidth, and low-crosstalk. r 2009 Elsevier GmbH. All rights reserved. Keywords: Optical coupler; Adiabatic directional coupler; Mismatched optical coupler; Coupling waveguide structure; Hamming function
1. Introduction Optical couplers are fundamental components of several useful optical waveguide devices including switches, modulators, and wavelength filters. These devices are required to have the characteristics of wavelength-independent and much lower level of crosstalk. To meet this demand for optical coupler, the mismatched optical directional coupler (MODC) is a suitable choice to features the transferring power from one guide to the other and establishing an adiabatic coupling in the core layer. The MODC consists of two non-uniform mismatched waveguides and the waveCorresponding author. Fax: +886 3 4254501.
E-mail address: [email protected] (C.-F. Chen). 0030-4026/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.09.028
guide structure change is slow enough to conform to adiabatic principles which is one that is sufficiently slow in varying so that there are on transitions between energy levels and hence no changes in level populations as the system is changed from one stage to another. It is also called adiabatic optical coupler. It is extended by adiabatic directional coupler (ADC). The ADC was proposed and theoretically analyzed in microwave applications [1–3]. A mathematical mechanical analog of an adiabatic coupler had been investigated [1]. Several adiabatic couplers such as 3 dB and full couplers had been suggested [2]. A more detailed analysis based on adiabatic principles had been shown [3]. It has advantage of low crosstalk. However, due to the microwave ranges between millimeter and meter, it seems impractical for such coupler having the length of
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several meters to satisfy the requirement of 5–10 times of coupling length of the parallel transmission lines. Until the application of this coupler extends from microwave to optical wave, the coupler that has the length of several millimeters becomes reasonable and workable in the real application. The first adiabatic optical coupler was investigated [4] as a means to achieve improvement in the design tolerance for a ‘‘crossed-beta’’ coupler. An analytic solution of the ‘‘crossed-beta’’ coupler was subsequently derived under the constraints of linear variation of propagation constants and constant coefficient [5,6]. These works give a remarkably simple expression for the coupler efficiency which applied only to full couplers with long path lengths. Several weighting functions were applied to MODC proceed with simulation and fabrication [7–10]. The variance of waveguide structures of MODC were constrained by these weighting functions. The results demonstrate that the efficiency for the constant width of waveguide structure (CWWS) is obviously influenced by different weighting functions. Among the weighting functions, the Hamming function proved to have the preferable performances for MODC. A crosstalk of 28 dB has obtained for the coupling strength is carefully weighted in the fabricated MODC with titanium-diffused waveguide by Hamming weighting function [7]. Significant improvement in sidelobe level of 29.16 dB and coupling length of 6.2 mm are achieved in a tapered velocity coupler weighted by Hamming function [8]. The MODC with the variable width of waveguide structure (VWWS) was proposed and optimally designed [9,10]. Utilizing the sin-square and raised-cosine weighting functions on the VWWS of MODC, the coupler lengths/bandwidths were end up to be 6.25 mm/ 290 nm and 5.8 mm/220 nm, respectively, under the crosstalk of 35 dB. In addition, the criteria for adiabatic power flow along tapered mismatched directional couplers were presented [11]. Conditions for which an adiabatic coupler could be used to obtain approximately 100% power transfer from one core to the other were given. The analytical solutions of optical directional couplers with linearly tapered waveguide structures were proposed, and explicitly quantitative design rules for determining the required length for a given performance tolerance were also determined [12]. In this paper, we adopted the coupled mode equations established previously for the tapered coupler. After then, we used the Fourier transform to express the dynamic power flow in the domain of normalized propagation distance in order for the Hamming weighting function to be applied and integrated to obtain the crosstalk. We investigated the MODC with two different waveguide structures of CWWS and VWWS weighted by the Hamming function. The optimum waveguide
structure parameters are obtained by the global search algorithm and beam propagation method (BPM). The performances of coupler length, bandwidth, and crosstalk for the CWWS and VWWS are studied and compared in detail in the discussions. It is shown that the waveguide structure parameters are crucial in determining the required optical properties of MODC.
2. Theoretical model The MODC consisting of two waveguides are shown in Fig. 1(b). The power launches into guide 1 at z ¼ 0 and travels along in the z direction. The coupled mode equations and Fourier transform method established previously for the tapered coupler was adopted for analyzing MODC with variable widths of waveguide [3,13]. For VWWS, the powers in the two guides are expressed as [3] P1 ðzÞ ¼ jE 1 ðzÞj2 xðzÞ xðzÞ jw1 ðzÞj2 þ sin2 jw2 ðzÞj2 ¼ cos2 2 2 sinðxðzÞÞ Reðw1 ðzÞw2 ðzÞÞ,
Fig. 1. Schematic diagram of the MODC with (a) CWWS and (b) VWWS.
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P2 ðzÞ ¼ jE 2 ðzÞj2 2 xðzÞ 2 2 xðzÞ ¼ sin jw1 ðzÞj þ cos jw2 ðzÞj2 2 2 þ sinðxðzÞÞ Reðw1 ðzÞw2 ðzÞÞ,
(1)
where E 1;2 ðzÞ are the wave amplitudes in guide 1 and 2, and w1;2 ðzÞ are the in-phase quasi-normal and out-phase quasi-normal modes, respectively. The function w2 ðzÞ represents the complex conjugate of w2(z) and Reðw1 ðzÞw2 ðzÞÞ indicates the real part of w1 ðzÞw2 ðzÞ. Several symbols being used in the Eq. (1) is defined in [3]: bðzÞ ¼ 12½b1 ðzÞ þ b2 ðzÞ,
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Let xðzÞ ¼ 0 at z ¼ 0 in Eq. (1), the powers at the input ends can be expressed as P1 ð0Þ ¼ jw1 ð0Þj2 and P2 ð0Þ ¼ jw2 ð0Þj2 , respectively. Assume the power initially exists at the input end in guide 1 such that P1(0) ¼ 1 and P2(0) ¼ 0. Thus only one of the quasi-normal modes will be excited under such conditions, then w1(0) ¼ 1 and w2(0) ¼ 0. The dynamic powers flow in the two guides can be written as P1 ðzÞ ffi cos2 ð12xðzÞÞf1 þ vðzÞg þ sin2 ð12xðzÞÞmðzÞ þ sinðxðzÞÞdðzÞ, P2 ðzÞ ffi sin2 ð12xðzÞÞf1 þ vðzÞg þ cos2 ð12xðzÞÞmðzÞ sinðxðzÞÞdðzÞ,
(4)
where
jðzÞ ¼ 12½b2 ðzÞ b1 ðzÞ,
Z 2 Z z 2 R z0 1 z dx 2jrðz0 Þ 0 j ðb1 b2 Þ dz00 0 e dz ¼ c e mðzÞ ¼ 12 , 4 0 dz0 0
jðzÞ cot xðzÞ ¼ , cðzÞ
Z
Z
z0
dx 2jrðz00 Þ 00 0 e dz dz , dz00
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ j2 ðzÞ þ c2 ðzÞ, GðzÞ ¼ lb0 ðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðzÞ GðzÞ þ jðzÞ cos ¼ , 2 2GðzÞ
1 vðzÞ ¼ Re 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðzÞ GðzÞ jðzÞ sin ¼ , 2 2GðzÞ
Let mðzÞ þ vðzÞ ¼ 0 to satisfy the energy conservation and xðlÞ ¼ p for a coupler substituting in Eqs. (1) and (4), the power in following is obtained:
dx 2jrðz0 Þ e dz0
0
Z z 1 dx 2jrðz0 Þ 0 2jrðzÞ dðzÞ ¼ Re e e dz . 0 2 0 dz
(2)
where b1;2 ðzÞ are uncoupled quasi-phase constants of guides 1 and 2, respectively, c(z) is the mutual and selfcoupling coefficient between the guides, cot xðzÞ is the ratio of variation of phase constant to variation of coupling coefficient, and lb0 ðzÞ is the local beat wavelength. For symmetry, it is assumed that b ¼ constant and jð0ÞX0. The traveling wave in the opposite direction is neglected in considering that j(z) and c(z) vary slowly compared to lb0 ðzÞ. And w1(z) and w2(z) are approximately given by [3] Z z 1 dx 2jrðz0 Þ 0 e dz w1 ðzÞ ffi ejrðzÞ w1 ð0Þ þ w2 ð0Þ 0 2 0 dz Z 0 Z z 1 dx 2jrðz0 Þ z dx 2jrðz00 Þ 00 0 w1 ð0Þ e e dz dz , 0 00 4 0 dz 0 dz Z z 1 dx 2jrðz0 Þ 0 w2 ðzÞ ffi ejrðzÞ w2 ð0Þ w1 ð0Þ e dz 0 2 dz 0 Z z Z 0 1 dx 2jrðz0 Þ z dx 2jrðz00 Þ 00 0 w2 ð0Þ e e dz dz , 0 00 4 0 dz 0 dz Rz
0
z
(3) 2
2
where rðzÞ ¼ 0 Gðz0 Þ dz0 . jw1 ð0Þj and jw2 ð0Þj represent the amounts of power excited in the in-phase quasinormal mode and out-phase quasi-normal mode, respectively. The amplitudes are normalized so that jw1 ð0Þj2 þ jw2 ð0Þj2 ¼ 1.
(5)
P1 ðlÞ ¼ jw2 ðlÞj2 ¼ mðlÞ, P2 ðlÞ ¼ jw1 ðlÞj2 ¼ 1 þ vðlÞ.
(6)
In Eq. (6), m(l) represents a residual power at the output end in guide 1. It is appropriate to term it as the ‘‘mode crosstalk’’. Since the input power in guides of the coupler is redistributed through the almost unchanged quasi-normal mode, this kind of coupler is known as an adiabatic directional coupler. In addition, we can express the mode crosstalk m(l) in Fourier transform as [13] Z 2 1 l dx 2jrðz0 Þ 0 e dz mðlÞ ¼ 4 0 dz0 Z l 2 R s0 1 dx j2l Gðs00 Þ ds00 0 e 0 ¼ ðl ds Þ 0 4 0 l ds Z 2 1 l dx jul 0 ¼ e ds 4 0 ds0 Z 2 1 g dx iul e du ; ¼ (7) 4 0 du R s0 where u ¼ 2 0 Gðs00 Þ ds00 , g ¼ uðlÞ. The crosstalk related to the coupling length l is proportional to the square of the Fourier transform of dx=du. In this study, we are interested in the constant local beat wavelength of coupler, therefore GðzÞ ¼
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 ðzÞ þ c2 ðzÞ ¼ p=lb0 ¼ constant. The crosstalk formula of Eq. (7) can be rewritten as Z Z 2 1 z dx 2jrðz0 Þ 0 2 1 k dx jð2pl=lb0 Þk0 0 mðk; DÞ ¼ e dz ¼ e dk 4 0 dz0 4 0 dk0 Z k 2 0 1 ¼ f ðk0 Þ ej2pDk dk0 , (8) 4 0 where tapered function f ðk0 Þ ¼ dx=dk0 , normalized propagation distance k ¼ z=l, and normalized coupler length D ¼ l=lb0 . Due to the coupling coefficient equal to zero at normalized propagation distances z/l ¼ 0 and z/l ¼ 1, there will exist infinite gap parameter G2 at these points [8]. In order to obtain a practical device, we introduce the truncation range of 0:05pz=lp0:95 to do the simulations. One can see that it resembles the Fourier transform with k ¼ 0.95 and f(k) is zero outside the range of 0:05pkp0:95. On the other hand, the crosstalk at the ends of guides m(1,D) represents the frequency domain of f(k). To study the crosstalk of MODC, we select the Hamming weighting function [7,8] f(k) written in the following: f ðkÞ ¼ 1 0:852 cosð2pkÞ.
(9)
Eqs. (8) and (9) are generally used to study the crosstalk of MODC. To understand the effect of operation wavelength in this coupler, the theoretical calculations of the variations of crosstalk as a function of coupler length for different wavelengths are shown in Fig. 2. The results show that the crosstalk periodically varies with the coupler length and the local maxima gradually decrease with the increasing coupler length. As the crosstalk of 35 dB is considered at the operation wavelengths of 1.53, 1.565, and 1.61 mm, the coupler lengths of MOC
Fig. 2. Variations of crosstalk as a function of coupler length for the theoretical calculations. (Solid, dashed, and dotted lines represent for the operation wavelength of 1.61, 1.565, and 1.53 mm, respectively.)
are 4.24, 4.36, and 4.46 mm, respectively. Under the consideration of C, L, and C+L band, the crosstalk will be deteriorated to a value of 46, 44.5, and 39 dB corresponding to a coupler length of 4.41, 4.29, and 4.33 mm, respectively, as shown in inset of Fig. 2. This result shows variation of crosstalk that brings drift in virtue of the effect of operation wavelength for a correlated coupling length. It is worth noting that the variations in crosstalk are related to the operation wavelength. Therefore, when the demanded C+L-broad is a necessary condition, the range of coupler length must have a sufficient range. However, the analytic theory of MODC is based on the adiabatic principles; in other words, when the coupling length of an adiabatic coupler is not long enough, the validity of the approximation theory becomes questionable [3,10]. In addition, theoretical results have not considered the practical waveguide structure parameters cannot be fabricated, so these results can only be regarded as referral. It is obvious that the effect for coupling waveguide structure of the adiabatic coupler cannot be permanently ignored. In particular, when the coupling length of the adiabatic coupler is not long enough, such an effect is more serious. Therefore, it is necessary for the numerical analysis of coupling waveguide structure of such coupler.
3. Results and discussions Due to the lack of structure parameters of the waveguide, one cannot fabricate a practical MODC based on the theoretical calculations. The purpose of this work is to implement the global search algorithm and BPM simulations to obtain the optimal waveguide structure parameters for MODC with the characteristics of short length, broad bandwidth, and low crosstalk. The two schematic diagrams of designed waveguide structures of variable widths and constant width are shown in Fig. 1(a) and (b), respectively. In Fig. 1(a), G1 and G2 are the center distances between the two waveguides at z ¼ L/2 and z ¼ 0, and guides 1 and 2 have the uniform width of W1. In Fig. 1(b), W1 and W2 are the widths of guides 1 and 2 at z ¼ L, respectively. In BPM simulation, Fig. 3(a) and (b) shows the power evolution of MODC with CWWS and VWWS along the waveguides, respectively. The mode power launches from left guide 1 and gradually switches into right guide 2. For the CWWS type in Fig. 3(a), the mode crosstalk must be very large due to the presence of a lobe of residual power at the output end in guide 1. However, for the VWWS type in Fig. 3(b), it can be easily seen that almost the flat level of residual power at the output end in guide 1 and the mode crosstalk VWWS type is
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Fig. 3. BPM simulation propagations of power evolutions of MODC with (a) CWWS and (b) VWWS.
clearly smaller than the CWWS type. The data of the power evolution are recorded with this launch configuration and repeated with different coupler lengths. The crosstalk is defined as 10 times the logarithm of the residual power in guide 1 divided by the input power. The parameters being used in the numerical study of MODC are: the waveguide core index nco ¼ 1:544, the waveguide cladding index ncl ¼ 1:523, and minimum local beat length lb0 ¼ 2000 mm at operation wavelength of 1.57 mm. We then used the global search algorithm and BPM simulations to obtain the suitable structure parameters of MODC for the demand of short-length, low-crosstalk, and bandwidths within C- or L-band and C+L-band. Here we wish that the demanded crosstalk (DC) condition is 35 dB. The results and discussions are shown in the following sections.
3.1. BPM simulations of CWWS for C- or L-band According to the conditions of shorter coupler length, crosstalk 35 dB, and bandwidths within C- or L-band, we analyzed the simulation results of MODC with CWWS and found that it was impossible. Therefore, we reduced the demand of crosstalk from 35 to 20 dB. A set of suitable structure parameter are obtained: G1 ¼ 7.6 mm, G2 ¼ 14 mm, and W1 ¼ 4 mm and called sample-I. Fig. 4 shows the relations of (a) crosstalk as a function of coupler length, (b) wavelength bands with coupler length, (c) crosstalk as a function of wavelength ranges with different coupler lengths, and (d) normalized powers in the two guides as a function of normalized propagation distance for at operation wavelength
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Fig. 4. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) crosstalk vs. wavelength, and (d) normalized power vs. normalized propagation distance for the CWWS sample-I.
of 1.57 mm. In Fig. 4(a), sample-I has the coupler length of 4.5 mm. It was mentioned previously that the variations of crosstalk are related to the operation wavelength. Therefore, in order to satisfy the demand for C- or L-band, we must choose a suitable coupler length made within the sufficient ranges of coupler lengths. In considering the bandwidth requirement, we did BPM simulations and statistically collected the data to compare the relationships between bandwidth and coupler length that is shown in Fig. 4(b). In these figures, the white and black circles represent the lower and upper bound of wavelengths in a specified coupler length, respectively. It can be seen that sample-I has the ranges of coupler lengths within 3.9 and 4.4 mm to reach a bandwidth of 40 nm within the C- or L-band. Based on the ranges of coupler lengths being mentioned earlier, the simulations are preceded for the shortest coupler length for specified C- or L-band. Fig. 4(c) shows the variations of the crosstalk within the C- and L-band for sample-I with a coupler length of 4.4 and 3.9 mm, respectively. Fig. 4(d) shows the variations of the powers in the two guides as a function of normalized propagation distance along the waveguide propagation for sample-I with coupler length of 4.4 and 3.9 mm. We can see that the variation of the powers in sample-I is smoothly changed from guides 1 to 2. The power transfer is stable to ensure a low crosstalk and low attenuation for a given coupler length.
3.2. BPM simulations of VWWS for C- or L-band According to our target, we used the same methods to analyze the results of MODC with VWWS. The chosen
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Table 1. Designed structure parameters and coupler lengths for VWWS by BPM simulations when the demand of crosstalk and bandwidth within the C-, L-band or C+L-bands are fulfilled.. Waveguide type
Cases
Bandwidth
G1 (mm)
G2 (mm)
W1 (mm)
W2 (mm)
Band
Coupler length (mm)
VWWS (crosstalk: 35 dB)
Sample-II
C- or L-band (single band)
6.9
9.6
7.9
4.4
C
3.0
Sample-III
C+L-band (full band)
7.1
8.0
7.6
4.8
L C+L
2.6 3.6
order to understand the bandwidth of the two regions, we utilized coupler length of 6.0 mm to precede analytic wavelength with crosstalk shown in Fig. 5(c). It is obviously that the coupler length of 6.0 mm at DCcondition 35 dB is in the bandwidth of two regions between 1.34–1.41 and 1.54–1.70 mm. In order to acquire the shortest coupler length for specific C- or L-band, we used the ranges of coupler lengths obtained previously. Fig. 5(d) shows the relationship in wavelength with crosstalk within the C- and L-band for sample-II with a coupler length of 3.0 and 2.6 mm at DC-condition 35 dB, respectively. Furthermore, the shortest coupler length used to analyze power transference continue is shown in Fig. 5(e). The curve changed smoothly, and the power almost completely transferred from guides 1 to 2. According to the results, it can also be seen that the VWWS has the shorter coupler length and lower crosstalk than the CWWS within either C- or L-band. However, sample-II with longer coupler lengths between 6.1–6.2, 6.4–6.8, and 7.6–8.0 mm can be suitable for the C+L-band. In addition, we tried to shorten the coupler length within the C+L-band under the same method.
Fig. 5. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) wavelength vs. coupler length as the two regions, (d) crosstalk vs. wavelength and (e) normalized power vs. normalized propagation distance for the VWWS sample-II.
parameters and results are listed in Table 1. In Fig. 5(a), when the DC-condition 35 dB is considered 1.57 mm as the operation wavelength, optimal coupler length of sample-II is 2.4 mm. Fig. 5(b) indicates that sample-II may be in the range of coupler lengths between 2.6 and 3.2 mm in order to fulfill a bandwidth of 40 nm within the C- or L-band at DC-condition 35 dB. Furthermore, this figure reveals the ranges of coupler lengths between 6.0–6.2 and 7.4 mm has the bandwidth within the two regions. In
3.3. BPM simulations of VWWS for C+L-band Sample-III in Table 1 represents the sample of VWWS to fulfill the bandwidth requirement within the C+L band under the demanding conditions of crosstalk. Sample-III has the ability to cover the 80 nm of the bandwidth within the C+L-band at the coupler length of 3.6 mm. Fig. 6(a) shows the variations of the crosstalk as a function of coupler length for sample-III. Compare to sample-II in Fig. 5(a), it is apparent that the sample-III has met the requirement of C+L-band since it has a wider ranges of coupler length at crosstalk of 35 dB. The relationship between wavelength band and coupler length for sample-III at DC-condition 35 dB is shown in Fig. 6(b). In this figure, the white and black circles represent the lower and upper bound of wavelengths in a
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coupler length and demand of crosstalk. Although the theoretical condition can achieve the DC-condition of 35 dB, but its coupler length still remains long. In addition, due to the adiabatic principles and regardless of the actual structure parameters, the theory refers here can only be used as a reference guide. Even though adding the structure parameters in CWWS, one cannot retrieve a low crosstalk, short coupler length, and broad bandwidth within the C+L-band comparing to the theoretical calculations. By utilizing VWWS and optimum structure parameters, the coupler lengths of 3.0 and 2.6 mm are obtained to meet the demand of C- and L-band under a low crosstalk of 35 dB. After choosing the optimum structure parameters and coupler length of 3.6 mm, a broad bandwidth within the C+L-band can be captured at the same crosstalk of 35 dB.
4. Conclusion
Fig. 6. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) wavelength vs. coupler length as the two regions, (d) crosstalk vs. wavelength, and (e) normalized power vs. normalized propagation distance for the C+L-band VWWS sample-III.
specified coupler length, respectively. It can be easily found in Fig. 6(b) that the sample-III has a bandwidth of 80 nm within C+L-band in the ranges of coupler lengths within 3.6–4.3, 5.6–6.4, and 7.4 mm. At the same time, coupler length of 5.4 mm and the distance between 7.2 and 7.4 mm can altogether be extracted from the two regions of bandwidth. One can see in Fig. 6(c), the coupler length of 5.4 mm at DC-condition B can obtain bandwidth of two regions, which are 1.40–1.48 and 1.60–1.70 mm. After choosing the coupler length of 3.60 mm, the variation of crosstalk with the wavelength for sample-III is shown in Fig. 6(d). In this figure, we observed that sample-III can achieve a broad bandwidth of 80 nm within the C+L-band at the DC-condition 35 dB. Fig. 6(e) shows the variations of powers in the two guides as a function of normalized propagation distance for sample-III with a coupler length of 3.6 mm. We noticed that the curves of power transfer between guides 1 and 2 are also smooth enough to guarantee feasible results of low crosstalk and short coupler length. Consequently, structure parameters play an important role in determining the bandwidth under a chosen
The results demonstrated clearly that the designs of waveguide structure are necessary for short coupler length with broad bandwidth in MODC. In an adequate search of the waveguide parameters with G1 ¼ 7.1, G2 ¼ 8.0, W1 ¼ 7.6, and W2 ¼ 4.8 mm, which is MODC with VWWS in sample-III, we observe that the curves of the power transfer between the two coupling waveguides are smooth enough to guarantee for broad bandwidth of 80 nm of C+L-band with coupler length of 3.6 mm and crosstalk of 35 dB. Obviously, the selection and design of waveguide structure are very important to suit the qualities of a MODC.
Acknowledgment This work is partially supported by National Science Council, Republic of China, under Contract NSC-972221-E-008-019.
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Optik
Optics
Optik 121 (2010) 831–838 www.elsevier.de/ijleo
Short-length and broadband mismatched optical couplers with tapered Hamming function for C- and L-band Yun-Sheng Kua, Chi-Feng Chena,b,, Chang-Neng Shauoc, Hsiou-Jeng Shyc a
Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC Institute of Opto-Mechatronics Engineering, National Central University, Jhongli 32054, Taiwan, ROC c Materials & Electro-Optics Research Division, Chung Shan Institute of Science and Technology, P.O. Box 90008-8-8, Lung-Tan, Tao-Yuan, 325 Taiwan, ROC b
Received 5 May 2008; accepted 28 September 2008
Abstract Mismatched optical couplers with variable widths of waveguide tapered by Hamming function are numerically investigated in the demand of short-length, broadband, and low crosstalk. We used global search algorithm and beam propagation method to seek optimal structure parameters of coupling waveguide. The coupler length is 3.6 mm within the C+L-band (1.53–1.61 mm) for variable widths of waveguide at crosstalk level of 35 dB. Comparison with constant width of waveguide, the constant width of waveguide has a coupler length of 4.4 mm and can only achieve 20 dB of crosstalk within the C-band (1.53–1.565 mm). Obviously, the waveguide with variable widths has the advantage over constant width for the demand of short-length, bandwidth, and low-crosstalk. r 2009 Elsevier GmbH. All rights reserved. Keywords: Optical coupler; Adiabatic directional coupler; Mismatched optical coupler; Coupling waveguide structure; Hamming function
1. Introduction Optical couplers are fundamental components of several useful optical waveguide devices including switches, modulators, and wavelength filters. These devices are required to have the characteristics of wavelength-independent and much lower level of crosstalk. To meet this demand for optical coupler, the mismatched optical directional coupler (MODC) is a suitable choice to features the transferring power from one guide to the other and establishing an adiabatic coupling in the core layer. The MODC consists of two non-uniform mismatched waveguides and the waveCorresponding author. Fax: +886 3 4254501.
E-mail address: [email protected] (C.-F. Chen). 0030-4026/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.09.028
guide structure change is slow enough to conform to adiabatic principles which is one that is sufficiently slow in varying so that there are on transitions between energy levels and hence no changes in level populations as the system is changed from one stage to another. It is also called adiabatic optical coupler. It is extended by adiabatic directional coupler (ADC). The ADC was proposed and theoretically analyzed in microwave applications [1–3]. A mathematical mechanical analog of an adiabatic coupler had been investigated [1]. Several adiabatic couplers such as 3 dB and full couplers had been suggested [2]. A more detailed analysis based on adiabatic principles had been shown [3]. It has advantage of low crosstalk. However, due to the microwave ranges between millimeter and meter, it seems impractical for such coupler having the length of
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several meters to satisfy the requirement of 5–10 times of coupling length of the parallel transmission lines. Until the application of this coupler extends from microwave to optical wave, the coupler that has the length of several millimeters becomes reasonable and workable in the real application. The first adiabatic optical coupler was investigated [4] as a means to achieve improvement in the design tolerance for a ‘‘crossed-beta’’ coupler. An analytic solution of the ‘‘crossed-beta’’ coupler was subsequently derived under the constraints of linear variation of propagation constants and constant coefficient [5,6]. These works give a remarkably simple expression for the coupler efficiency which applied only to full couplers with long path lengths. Several weighting functions were applied to MODC proceed with simulation and fabrication [7–10]. The variance of waveguide structures of MODC were constrained by these weighting functions. The results demonstrate that the efficiency for the constant width of waveguide structure (CWWS) is obviously influenced by different weighting functions. Among the weighting functions, the Hamming function proved to have the preferable performances for MODC. A crosstalk of 28 dB has obtained for the coupling strength is carefully weighted in the fabricated MODC with titanium-diffused waveguide by Hamming weighting function [7]. Significant improvement in sidelobe level of 29.16 dB and coupling length of 6.2 mm are achieved in a tapered velocity coupler weighted by Hamming function [8]. The MODC with the variable width of waveguide structure (VWWS) was proposed and optimally designed [9,10]. Utilizing the sin-square and raised-cosine weighting functions on the VWWS of MODC, the coupler lengths/bandwidths were end up to be 6.25 mm/ 290 nm and 5.8 mm/220 nm, respectively, under the crosstalk of 35 dB. In addition, the criteria for adiabatic power flow along tapered mismatched directional couplers were presented [11]. Conditions for which an adiabatic coupler could be used to obtain approximately 100% power transfer from one core to the other were given. The analytical solutions of optical directional couplers with linearly tapered waveguide structures were proposed, and explicitly quantitative design rules for determining the required length for a given performance tolerance were also determined [12]. In this paper, we adopted the coupled mode equations established previously for the tapered coupler. After then, we used the Fourier transform to express the dynamic power flow in the domain of normalized propagation distance in order for the Hamming weighting function to be applied and integrated to obtain the crosstalk. We investigated the MODC with two different waveguide structures of CWWS and VWWS weighted by the Hamming function. The optimum waveguide
structure parameters are obtained by the global search algorithm and beam propagation method (BPM). The performances of coupler length, bandwidth, and crosstalk for the CWWS and VWWS are studied and compared in detail in the discussions. It is shown that the waveguide structure parameters are crucial in determining the required optical properties of MODC.
2. Theoretical model The MODC consisting of two waveguides are shown in Fig. 1(b). The power launches into guide 1 at z ¼ 0 and travels along in the z direction. The coupled mode equations and Fourier transform method established previously for the tapered coupler was adopted for analyzing MODC with variable widths of waveguide [3,13]. For VWWS, the powers in the two guides are expressed as [3] P1 ðzÞ ¼ jE 1 ðzÞj2 xðzÞ xðzÞ jw1 ðzÞj2 þ sin2 jw2 ðzÞj2 ¼ cos2 2 2 sinðxðzÞÞ Reðw1 ðzÞw2 ðzÞÞ,
Fig. 1. Schematic diagram of the MODC with (a) CWWS and (b) VWWS.
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P2 ðzÞ ¼ jE 2 ðzÞj2 2 xðzÞ 2 2 xðzÞ ¼ sin jw1 ðzÞj þ cos jw2 ðzÞj2 2 2 þ sinðxðzÞÞ Reðw1 ðzÞw2 ðzÞÞ,
(1)
where E 1;2 ðzÞ are the wave amplitudes in guide 1 and 2, and w1;2 ðzÞ are the in-phase quasi-normal and out-phase quasi-normal modes, respectively. The function w2 ðzÞ represents the complex conjugate of w2(z) and Reðw1 ðzÞw2 ðzÞÞ indicates the real part of w1 ðzÞw2 ðzÞ. Several symbols being used in the Eq. (1) is defined in [3]: bðzÞ ¼ 12½b1 ðzÞ þ b2 ðzÞ,
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Let xðzÞ ¼ 0 at z ¼ 0 in Eq. (1), the powers at the input ends can be expressed as P1 ð0Þ ¼ jw1 ð0Þj2 and P2 ð0Þ ¼ jw2 ð0Þj2 , respectively. Assume the power initially exists at the input end in guide 1 such that P1(0) ¼ 1 and P2(0) ¼ 0. Thus only one of the quasi-normal modes will be excited under such conditions, then w1(0) ¼ 1 and w2(0) ¼ 0. The dynamic powers flow in the two guides can be written as P1 ðzÞ ffi cos2 ð12xðzÞÞf1 þ vðzÞg þ sin2 ð12xðzÞÞmðzÞ þ sinðxðzÞÞdðzÞ, P2 ðzÞ ffi sin2 ð12xðzÞÞf1 þ vðzÞg þ cos2 ð12xðzÞÞmðzÞ sinðxðzÞÞdðzÞ,
(4)
where
jðzÞ ¼ 12½b2 ðzÞ b1 ðzÞ,
Z 2 Z z 2 R z0 1 z dx 2jrðz0 Þ 0 j ðb1 b2 Þ dz00 0 e dz ¼ c e mðzÞ ¼ 12 , 4 0 dz0 0
jðzÞ cot xðzÞ ¼ , cðzÞ
Z
Z
z0
dx 2jrðz00 Þ 00 0 e dz dz , dz00
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ j2 ðzÞ þ c2 ðzÞ, GðzÞ ¼ lb0 ðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðzÞ GðzÞ þ jðzÞ cos ¼ , 2 2GðzÞ
1 vðzÞ ¼ Re 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðzÞ GðzÞ jðzÞ sin ¼ , 2 2GðzÞ
Let mðzÞ þ vðzÞ ¼ 0 to satisfy the energy conservation and xðlÞ ¼ p for a coupler substituting in Eqs. (1) and (4), the power in following is obtained:
dx 2jrðz0 Þ e dz0
0
Z z 1 dx 2jrðz0 Þ 0 2jrðzÞ dðzÞ ¼ Re e e dz . 0 2 0 dz
(2)
where b1;2 ðzÞ are uncoupled quasi-phase constants of guides 1 and 2, respectively, c(z) is the mutual and selfcoupling coefficient between the guides, cot xðzÞ is the ratio of variation of phase constant to variation of coupling coefficient, and lb0 ðzÞ is the local beat wavelength. For symmetry, it is assumed that b ¼ constant and jð0ÞX0. The traveling wave in the opposite direction is neglected in considering that j(z) and c(z) vary slowly compared to lb0 ðzÞ. And w1(z) and w2(z) are approximately given by [3] Z z 1 dx 2jrðz0 Þ 0 e dz w1 ðzÞ ffi ejrðzÞ w1 ð0Þ þ w2 ð0Þ 0 2 0 dz Z 0 Z z 1 dx 2jrðz0 Þ z dx 2jrðz00 Þ 00 0 w1 ð0Þ e e dz dz , 0 00 4 0 dz 0 dz Z z 1 dx 2jrðz0 Þ 0 w2 ðzÞ ffi ejrðzÞ w2 ð0Þ w1 ð0Þ e dz 0 2 dz 0 Z z Z 0 1 dx 2jrðz0 Þ z dx 2jrðz00 Þ 00 0 w2 ð0Þ e e dz dz , 0 00 4 0 dz 0 dz Rz
0
z
(3) 2
2
where rðzÞ ¼ 0 Gðz0 Þ dz0 . jw1 ð0Þj and jw2 ð0Þj represent the amounts of power excited in the in-phase quasinormal mode and out-phase quasi-normal mode, respectively. The amplitudes are normalized so that jw1 ð0Þj2 þ jw2 ð0Þj2 ¼ 1.
(5)
P1 ðlÞ ¼ jw2 ðlÞj2 ¼ mðlÞ, P2 ðlÞ ¼ jw1 ðlÞj2 ¼ 1 þ vðlÞ.
(6)
In Eq. (6), m(l) represents a residual power at the output end in guide 1. It is appropriate to term it as the ‘‘mode crosstalk’’. Since the input power in guides of the coupler is redistributed through the almost unchanged quasi-normal mode, this kind of coupler is known as an adiabatic directional coupler. In addition, we can express the mode crosstalk m(l) in Fourier transform as [13] Z 2 1 l dx 2jrðz0 Þ 0 e dz mðlÞ ¼ 4 0 dz0 Z l 2 R s0 1 dx j2l Gðs00 Þ ds00 0 e 0 ¼ ðl ds Þ 0 4 0 l ds Z 2 1 l dx jul 0 ¼ e ds 4 0 ds0 Z 2 1 g dx iul e du ; ¼ (7) 4 0 du R s0 where u ¼ 2 0 Gðs00 Þ ds00 , g ¼ uðlÞ. The crosstalk related to the coupling length l is proportional to the square of the Fourier transform of dx=du. In this study, we are interested in the constant local beat wavelength of coupler, therefore GðzÞ ¼
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 ðzÞ þ c2 ðzÞ ¼ p=lb0 ¼ constant. The crosstalk formula of Eq. (7) can be rewritten as Z Z 2 1 z dx 2jrðz0 Þ 0 2 1 k dx jð2pl=lb0 Þk0 0 mðk; DÞ ¼ e dz ¼ e dk 4 0 dz0 4 0 dk0 Z k 2 0 1 ¼ f ðk0 Þ ej2pDk dk0 , (8) 4 0 where tapered function f ðk0 Þ ¼ dx=dk0 , normalized propagation distance k ¼ z=l, and normalized coupler length D ¼ l=lb0 . Due to the coupling coefficient equal to zero at normalized propagation distances z/l ¼ 0 and z/l ¼ 1, there will exist infinite gap parameter G2 at these points [8]. In order to obtain a practical device, we introduce the truncation range of 0:05pz=lp0:95 to do the simulations. One can see that it resembles the Fourier transform with k ¼ 0.95 and f(k) is zero outside the range of 0:05pkp0:95. On the other hand, the crosstalk at the ends of guides m(1,D) represents the frequency domain of f(k). To study the crosstalk of MODC, we select the Hamming weighting function [7,8] f(k) written in the following: f ðkÞ ¼ 1 0:852 cosð2pkÞ.
(9)
Eqs. (8) and (9) are generally used to study the crosstalk of MODC. To understand the effect of operation wavelength in this coupler, the theoretical calculations of the variations of crosstalk as a function of coupler length for different wavelengths are shown in Fig. 2. The results show that the crosstalk periodically varies with the coupler length and the local maxima gradually decrease with the increasing coupler length. As the crosstalk of 35 dB is considered at the operation wavelengths of 1.53, 1.565, and 1.61 mm, the coupler lengths of MOC
Fig. 2. Variations of crosstalk as a function of coupler length for the theoretical calculations. (Solid, dashed, and dotted lines represent for the operation wavelength of 1.61, 1.565, and 1.53 mm, respectively.)
are 4.24, 4.36, and 4.46 mm, respectively. Under the consideration of C, L, and C+L band, the crosstalk will be deteriorated to a value of 46, 44.5, and 39 dB corresponding to a coupler length of 4.41, 4.29, and 4.33 mm, respectively, as shown in inset of Fig. 2. This result shows variation of crosstalk that brings drift in virtue of the effect of operation wavelength for a correlated coupling length. It is worth noting that the variations in crosstalk are related to the operation wavelength. Therefore, when the demanded C+L-broad is a necessary condition, the range of coupler length must have a sufficient range. However, the analytic theory of MODC is based on the adiabatic principles; in other words, when the coupling length of an adiabatic coupler is not long enough, the validity of the approximation theory becomes questionable [3,10]. In addition, theoretical results have not considered the practical waveguide structure parameters cannot be fabricated, so these results can only be regarded as referral. It is obvious that the effect for coupling waveguide structure of the adiabatic coupler cannot be permanently ignored. In particular, when the coupling length of the adiabatic coupler is not long enough, such an effect is more serious. Therefore, it is necessary for the numerical analysis of coupling waveguide structure of such coupler.
3. Results and discussions Due to the lack of structure parameters of the waveguide, one cannot fabricate a practical MODC based on the theoretical calculations. The purpose of this work is to implement the global search algorithm and BPM simulations to obtain the optimal waveguide structure parameters for MODC with the characteristics of short length, broad bandwidth, and low crosstalk. The two schematic diagrams of designed waveguide structures of variable widths and constant width are shown in Fig. 1(a) and (b), respectively. In Fig. 1(a), G1 and G2 are the center distances between the two waveguides at z ¼ L/2 and z ¼ 0, and guides 1 and 2 have the uniform width of W1. In Fig. 1(b), W1 and W2 are the widths of guides 1 and 2 at z ¼ L, respectively. In BPM simulation, Fig. 3(a) and (b) shows the power evolution of MODC with CWWS and VWWS along the waveguides, respectively. The mode power launches from left guide 1 and gradually switches into right guide 2. For the CWWS type in Fig. 3(a), the mode crosstalk must be very large due to the presence of a lobe of residual power at the output end in guide 1. However, for the VWWS type in Fig. 3(b), it can be easily seen that almost the flat level of residual power at the output end in guide 1 and the mode crosstalk VWWS type is
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Fig. 3. BPM simulation propagations of power evolutions of MODC with (a) CWWS and (b) VWWS.
clearly smaller than the CWWS type. The data of the power evolution are recorded with this launch configuration and repeated with different coupler lengths. The crosstalk is defined as 10 times the logarithm of the residual power in guide 1 divided by the input power. The parameters being used in the numerical study of MODC are: the waveguide core index nco ¼ 1:544, the waveguide cladding index ncl ¼ 1:523, and minimum local beat length lb0 ¼ 2000 mm at operation wavelength of 1.57 mm. We then used the global search algorithm and BPM simulations to obtain the suitable structure parameters of MODC for the demand of short-length, low-crosstalk, and bandwidths within C- or L-band and C+L-band. Here we wish that the demanded crosstalk (DC) condition is 35 dB. The results and discussions are shown in the following sections.
3.1. BPM simulations of CWWS for C- or L-band According to the conditions of shorter coupler length, crosstalk 35 dB, and bandwidths within C- or L-band, we analyzed the simulation results of MODC with CWWS and found that it was impossible. Therefore, we reduced the demand of crosstalk from 35 to 20 dB. A set of suitable structure parameter are obtained: G1 ¼ 7.6 mm, G2 ¼ 14 mm, and W1 ¼ 4 mm and called sample-I. Fig. 4 shows the relations of (a) crosstalk as a function of coupler length, (b) wavelength bands with coupler length, (c) crosstalk as a function of wavelength ranges with different coupler lengths, and (d) normalized powers in the two guides as a function of normalized propagation distance for at operation wavelength
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Fig. 4. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) crosstalk vs. wavelength, and (d) normalized power vs. normalized propagation distance for the CWWS sample-I.
of 1.57 mm. In Fig. 4(a), sample-I has the coupler length of 4.5 mm. It was mentioned previously that the variations of crosstalk are related to the operation wavelength. Therefore, in order to satisfy the demand for C- or L-band, we must choose a suitable coupler length made within the sufficient ranges of coupler lengths. In considering the bandwidth requirement, we did BPM simulations and statistically collected the data to compare the relationships between bandwidth and coupler length that is shown in Fig. 4(b). In these figures, the white and black circles represent the lower and upper bound of wavelengths in a specified coupler length, respectively. It can be seen that sample-I has the ranges of coupler lengths within 3.9 and 4.4 mm to reach a bandwidth of 40 nm within the C- or L-band. Based on the ranges of coupler lengths being mentioned earlier, the simulations are preceded for the shortest coupler length for specified C- or L-band. Fig. 4(c) shows the variations of the crosstalk within the C- and L-band for sample-I with a coupler length of 4.4 and 3.9 mm, respectively. Fig. 4(d) shows the variations of the powers in the two guides as a function of normalized propagation distance along the waveguide propagation for sample-I with coupler length of 4.4 and 3.9 mm. We can see that the variation of the powers in sample-I is smoothly changed from guides 1 to 2. The power transfer is stable to ensure a low crosstalk and low attenuation for a given coupler length.
3.2. BPM simulations of VWWS for C- or L-band According to our target, we used the same methods to analyze the results of MODC with VWWS. The chosen
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Table 1. Designed structure parameters and coupler lengths for VWWS by BPM simulations when the demand of crosstalk and bandwidth within the C-, L-band or C+L-bands are fulfilled.. Waveguide type
Cases
Bandwidth
G1 (mm)
G2 (mm)
W1 (mm)
W2 (mm)
Band
Coupler length (mm)
VWWS (crosstalk: 35 dB)
Sample-II
C- or L-band (single band)
6.9
9.6
7.9
4.4
C
3.0
Sample-III
C+L-band (full band)
7.1
8.0
7.6
4.8
L C+L
2.6 3.6
order to understand the bandwidth of the two regions, we utilized coupler length of 6.0 mm to precede analytic wavelength with crosstalk shown in Fig. 5(c). It is obviously that the coupler length of 6.0 mm at DCcondition 35 dB is in the bandwidth of two regions between 1.34–1.41 and 1.54–1.70 mm. In order to acquire the shortest coupler length for specific C- or L-band, we used the ranges of coupler lengths obtained previously. Fig. 5(d) shows the relationship in wavelength with crosstalk within the C- and L-band for sample-II with a coupler length of 3.0 and 2.6 mm at DC-condition 35 dB, respectively. Furthermore, the shortest coupler length used to analyze power transference continue is shown in Fig. 5(e). The curve changed smoothly, and the power almost completely transferred from guides 1 to 2. According to the results, it can also be seen that the VWWS has the shorter coupler length and lower crosstalk than the CWWS within either C- or L-band. However, sample-II with longer coupler lengths between 6.1–6.2, 6.4–6.8, and 7.6–8.0 mm can be suitable for the C+L-band. In addition, we tried to shorten the coupler length within the C+L-band under the same method.
Fig. 5. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) wavelength vs. coupler length as the two regions, (d) crosstalk vs. wavelength and (e) normalized power vs. normalized propagation distance for the VWWS sample-II.
parameters and results are listed in Table 1. In Fig. 5(a), when the DC-condition 35 dB is considered 1.57 mm as the operation wavelength, optimal coupler length of sample-II is 2.4 mm. Fig. 5(b) indicates that sample-II may be in the range of coupler lengths between 2.6 and 3.2 mm in order to fulfill a bandwidth of 40 nm within the C- or L-band at DC-condition 35 dB. Furthermore, this figure reveals the ranges of coupler lengths between 6.0–6.2 and 7.4 mm has the bandwidth within the two regions. In
3.3. BPM simulations of VWWS for C+L-band Sample-III in Table 1 represents the sample of VWWS to fulfill the bandwidth requirement within the C+L band under the demanding conditions of crosstalk. Sample-III has the ability to cover the 80 nm of the bandwidth within the C+L-band at the coupler length of 3.6 mm. Fig. 6(a) shows the variations of the crosstalk as a function of coupler length for sample-III. Compare to sample-II in Fig. 5(a), it is apparent that the sample-III has met the requirement of C+L-band since it has a wider ranges of coupler length at crosstalk of 35 dB. The relationship between wavelength band and coupler length for sample-III at DC-condition 35 dB is shown in Fig. 6(b). In this figure, the white and black circles represent the lower and upper bound of wavelengths in a
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coupler length and demand of crosstalk. Although the theoretical condition can achieve the DC-condition of 35 dB, but its coupler length still remains long. In addition, due to the adiabatic principles and regardless of the actual structure parameters, the theory refers here can only be used as a reference guide. Even though adding the structure parameters in CWWS, one cannot retrieve a low crosstalk, short coupler length, and broad bandwidth within the C+L-band comparing to the theoretical calculations. By utilizing VWWS and optimum structure parameters, the coupler lengths of 3.0 and 2.6 mm are obtained to meet the demand of C- and L-band under a low crosstalk of 35 dB. After choosing the optimum structure parameters and coupler length of 3.6 mm, a broad bandwidth within the C+L-band can be captured at the same crosstalk of 35 dB.
4. Conclusion
Fig. 6. Relations of (a) crosstalk vs. coupler length, (b) wavelength band vs. coupler length, (c) wavelength vs. coupler length as the two regions, (d) crosstalk vs. wavelength, and (e) normalized power vs. normalized propagation distance for the C+L-band VWWS sample-III.
specified coupler length, respectively. It can be easily found in Fig. 6(b) that the sample-III has a bandwidth of 80 nm within C+L-band in the ranges of coupler lengths within 3.6–4.3, 5.6–6.4, and 7.4 mm. At the same time, coupler length of 5.4 mm and the distance between 7.2 and 7.4 mm can altogether be extracted from the two regions of bandwidth. One can see in Fig. 6(c), the coupler length of 5.4 mm at DC-condition B can obtain bandwidth of two regions, which are 1.40–1.48 and 1.60–1.70 mm. After choosing the coupler length of 3.60 mm, the variation of crosstalk with the wavelength for sample-III is shown in Fig. 6(d). In this figure, we observed that sample-III can achieve a broad bandwidth of 80 nm within the C+L-band at the DC-condition 35 dB. Fig. 6(e) shows the variations of powers in the two guides as a function of normalized propagation distance for sample-III with a coupler length of 3.6 mm. We noticed that the curves of power transfer between guides 1 and 2 are also smooth enough to guarantee feasible results of low crosstalk and short coupler length. Consequently, structure parameters play an important role in determining the bandwidth under a chosen
The results demonstrated clearly that the designs of waveguide structure are necessary for short coupler length with broad bandwidth in MODC. In an adequate search of the waveguide parameters with G1 ¼ 7.1, G2 ¼ 8.0, W1 ¼ 7.6, and W2 ¼ 4.8 mm, which is MODC with VWWS in sample-III, we observe that the curves of the power transfer between the two coupling waveguides are smooth enough to guarantee for broad bandwidth of 80 nm of C+L-band with coupler length of 3.6 mm and crosstalk of 35 dB. Obviously, the selection and design of waveguide structure are very important to suit the qualities of a MODC.
Acknowledgment This work is partially supported by National Science Council, Republic of China, under Contract NSC-972221-E-008-019.
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