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Design of short length and C+L-band mismatched optical coupler with waveguide weighted by the Blackman function
Optics Communications 282 (2009) 208–213
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of short length and C+L-band mismatched optical coupler with waveguide weighted by the Blackman function Chi-Feng Chen a,*, Yun-Sheng Ku b, Tsu-Te Kung c a
Institute of Opto-Mechatronics Engineering/Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC c Department of Electro-Optical Engineering, National United University, Miaoli 36003, Taiwan, ROC b
a r t i c l e
i n f o
Article history: Received 26 March 2008 Received in revised form 1 October 2008 Accepted 1 October 2008
Keywords: Optical coupler Adiabatic directional couplers Mismatched optical coupler Coupling waveguide structure Blackman weighting function
a b s t r a c t A mismatched optical coupler with waveguide weighted by the Blackman function is numerically investigated in the demand of short length, C+L-band, and low crosstalk. Utilizing the full factorial design, the structure parameters of coupling waveguide are obtained by beam propagation method. In the condition of crosstalk of 35 dB, the mismatched optical coupler with proper selected waveguide structure parameters is found to have a coupling length of 3.60 mm in the transmission wavelength ranges of C+L-band (1.53–1.61 lm). Obviously, the selection and design of waveguide structure are very important to satisfy the qualities of a mismatched optical coupler for the demand of short length, broad bandwidth, and low crosstalk. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Optical couplers are the 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 with switching capability, the mismatched optical coupler (MOC) is a suitable choice to have the features of transferring power from one guide to the other and establishing an adiabatic coupling in the core layer. Previously, the MOC was proposed and theoretically analyzed in the microwave applications [1–3]. Due to the microwave ranges between millimeter and meter, it seems impractical for such coupler having the length of 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 which has 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 * Corresponding author. Tel.: +886 3 4267308; fax: +886 3 4254501. E-mail address: [email protected] (C.-F. Chen). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.012
efficiency which applied only to full couplers with long path lengths. Several weighting functions were applied to the MOC to proceed with the simulation and fabrication [7–9]. The variable width of waveguide structures (VWWS) of MOC were constrained by these weighting functions. The results demonstrate that the efficiency for the VWWS is obviously influenced by the different weighting functions. Among the weighting functions, the Blackman function was proved to have the preferable performances for a MOC. Significant improvement in crosstalk level of 33.67 dB and coupling length of 8.2 mm were also obtained in a tapered velocity coupler weighted by Blackman function [7]. The MOC with the variable width of waveguide structure was proposed and optimally designed [8,9]. Utilizing the sin-square and raised-cosine weighting functions on the MOC, the coupling length/bandwidths were obtained to be 6.25 mm/290 nm and 5.8 mm/220 nm, respectively under the crosstalk of 35 dB. In addition, the criteria of adiabatic power flow along tapered mismatched directional couplers were presented [10]. 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 [11]. In this paper, we adopt the coupled mode equations established previously for the tapered coupler. After then, we use the Fourier
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transform to express the dynamic power flow in the domain of normalized propagation distance in order for the Blackman weighting function to be applied and integrated to obtain the crosstalk. We investigate the MOC with VWWS weighted by the Blackman function. The optimum waveguide structure parameters are obtained by the beam propagation method (BPM) [8,9,11] of utilizing the full factorial method [12]. The performances of coupler length, crosstalk, process tolerances, and C+L-band for the MOC are studied in detail through the discussions. It is shown that the waveguide structure parameters are crucial in determining the required optical properties of MOC. 2. Theoretical model The MOC consists of two waveguides as shown in Fig. 1. The power launches into guide 1 at z = 0 and travels along in the z direction. We focus on the forward normal modes in the co-directional coupler and ignore the backward normal modes. The following equations govern the two forward normal modes for such a waveguide system. The wave amplitudes for the two coupled waveguides can be written in the following:
dA1 ðzÞ ¼ j½b1 ðzÞ þ jðzÞA1 ðzÞ þ jjðzÞA2 ðzÞ; dz dA2 ðzÞ ¼ jjðzÞA1 ðzÞ j½b2 ðzÞ þ jðzÞA2 ðzÞ; dz
ð1Þ
where A1,2(z) are wave amplitudes in guide 1 and 2, respectively, b1,2(z) are uncoupled local-phase constants of guides 1 and 2, respectively. The mutual and self-coupling coefficients j(z) are as-
sumed to be identical. j(z) also must be a real value to satisfy the conservation of energy. The wave amplitudes are normalized so that the power in either guide is equal to the square of the wave amplitudes. The phase constants of the two composite guides vary in the reverse direction and cross over at some points of the coupler. Several symbols being used in the tapered power equations are defined as [3]
bðzÞ ¼ 1=2½b1 ðzÞ þ b2 ðzÞ; dðzÞ ¼ 1=2½b2 ðzÞ b1 ðzÞ; dðzÞ cot nðzÞ ¼ ; jðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p CðzÞ ¼ ¼ d2 ðzÞ þ j2 ðzÞ ¼ jðzÞ 1 þ M2 ; kb0 ðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðzÞ CðzÞ þ dðzÞ cos ¼ ; 2 2CðzÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s nðzÞ CðzÞ dðzÞ ; sin 2 2CðzÞ
ð2Þ
where M is termed the mismatch value that measures the synchronicity of the guides, cot n(z) is the ratio of variation of phase constant to variation of coupling coefficient, and kb0 ðzÞ is the minimum local beat length. For symmetry, b = constant and d(0) P 0. It is considered that d(z) and j(z) vary slowly compared to kb0 ðzÞ. And the traveling wave in the opposite direction is neglected. The characteristic impedances are normalized so that the power in either guide is equal to the square of the wave amplitude. The powers of single tapered mode directional couplers in the two guides are expressed as [3]
P1 ðzÞ ¼ jA1 ðzÞj2 ¼ cos2
nðzÞ 2 nðzÞ jN A ðzÞj2 sin jNB ðzÞj2 2 2
sinðnðzÞÞReðNA ðzÞNB ðzÞÞ; nðzÞ 2 nðzÞ jN A ðzÞj2 cos2 jNB ðzÞj2 P2 ðzÞ ¼ jA2 ðzÞj2 ¼ sin 2 2
ð3Þ
sinðnðzÞÞReðNA ðzÞNB ðzÞÞ; where A1,2(z) are the wave amplitudes in guide 1 and 2, and NA,B(z) are the in-phase local-normal and out-phase local-normal modes, respectively. The function NB ðzÞ represents the complex conjugate of NB(z) and ReðN A ðzÞN B ðzÞÞ indicates the real part of N A ðzÞN B ðzÞ. And NA(z) and NB(z) are approximately given by [3]
Z z 1 dn 2jqðz0 Þ 0 NA ðzÞ ffi ejqðzÞ N A ð0Þ þ N B ð0Þ dz 0e 2 0 dz ! Z 0 Z z 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 NA ð0Þ dz dz ; 0e 00 e 4 0 dz 0 dz Z z 1 dn 2jqðz0 Þ 0 NB ðzÞ ffi ejqðzÞ NB ð0Þ N A ð0Þ dz 0e 2 0 dz ! Z z Z 0 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 dz dz ; NB ð0Þ 0e 00 e 4 dz 0 0 dz
Fig. 1. Schematics diagrams of the MOC with three-dimensional variable widths of waveguide structure viewed in the (a) plan surface of X–Z and (b) cross-section of X–Y.
ð4Þ
Rz 0 where qðzÞ ¼ 0 Cðz0 Þdz jN A ð0Þj2 and jNB ð0Þj2 represent the amounts of power excited in the in-phase local-normal mode and out-phase local-normal mode, respectively. The amplitudes are normalized so that jNA ð0Þj2 þ jN B ð0Þj2 ¼ 1. Let n(z) = 0 at z = 0 in Eq. (1), the powers at the input ends can be expressed as P1 ð0Þ ¼ jN A ð0Þj2 and P 2 ð0Þ ¼ jN B ð0Þj2 , respectively. Assume the power initially exists at the input end in guide 1 such that P1(0) = 1 and P2(0) = 0. Thus only one of the local-normal modes will be excited under such conditions, then NA(0) = 1 and NB(0) = 0. The dynamic power flows in the two guides can be written as
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1 2 1 P1 ðzÞ ffi cos2 nðzÞ f1 þ mðzÞg þ sin nðzÞ lðzÞ þ sinðnðzÞÞDðzÞ; 2 2 2 1 2 1 P2 ðzÞ ffi sin nðzÞ f1 þ mðzÞg þ cos nðzÞÞlðzÞ sinðnðzÞ DðzÞ; 2 2 ð5Þ where
2 Z 1 z dn 2jqðz0 Þ 0 lðzÞ ¼ e dz ; 4 0 dz0 ! Z 0 Z z 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 mðzÞ ¼ Re dz dz ¼ lðzÞ; 0e 00 e 2 dz 0 0 dz Z z 1 dn 2jqðz0 Þ 0 DðzÞ ¼ Re e2jqðzÞ dz : 0e 2 0 dz
ð6Þ
Let l(z) + v(z) = 0 to satisfy the energy conservation and n(l) = p for a coupler substitute into Eqs. (1) and (4) to obtain the powers in the following:
P1 ðlÞ ¼ jNB ðlÞj2 ¼ lðlÞ; P2 ðlÞ ¼ jNA ðlÞj2 ¼ 1 lðlÞ:
ð7Þ
In Eq. (6), l(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 local-normal mode, this kind of coupler is known as an adiabatic directional coupler. In addition, we can express the mode crosstalk lðlÞ in Fourier transform as [13]
Z 1 l lðlÞ ¼ 4 0 Z 1 l ¼ 4 0
2 2 Z R0 dn 2jqðz0 Þ 0 1 l dn j2l s 00 0 00 0 dz ¼ Cðs Þds ðlds Þ 0e 0 e 4 0 lds dz 2 Z 2 dn jul 0 1 g dn iul e du ; 0 e ds ¼ 4 0 du ds
ð8Þ
R s0 00 where u ¼ 2 0 Cðs00 Þds ; g ¼ uðlÞ. The crosstalk related to the coupling length l is proportional to the square of the Fourier transform dn .In this study, we are interested in the constant local beat of du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wavelength of coupler, therefore CðzÞ ¼ d2 ðzÞ þ j2 ðzÞ ¼ kp = conb0 stant. The crosstalk formula of Eq. (8) can be rewritten as
2 Z 2 dn 2jqðz0 Þ 0 1 k dn jk2pl 0 0 b0 k dk e dz ¼ e 4 0 dz0 4 0 dk0 2 Z 0 1 k 0 0 ¼ f ðk Þej2pDk dk ; 4 0 Z
1 lðk; DÞ ¼
z
ð9Þ
0
dn where tapered function f ðk Þ ¼ dk 0 , normalized propagation distance k ¼ zl, and normalized coupler length D ¼ k l . Due to the coupling b0 coefficient equal to zero at normalized propagation distances z/l = 0 and z/l = 1, there will exist infinite gap parameter G2 at these points [10]. In order to obtain a practical device, we introduce the truncation range of 0.05 6 z/l 6 0.95 to do the simulations [7]. One can see that it resembles the Fourier transform with k = 0.95 and f(k) is zero outside the range of 0.05 6 k 6 0.95. On the other hand, the crosstalk at the ends of guides l(1, D) represents the frequency domain of f(k). To study the crosstalk of MOC, we select the Blackman weighting function [7] f(k) written in the following:
f ðkÞ ¼ 1 1:19 cosð2pkÞ þ 0:19 cosð4pkÞ:
ð10Þ
Eqs. (9) and (10) are generally used to study the crosstalk of MOC. To understand the effect of operation wavelength in this coupling, the theoretical calculations of the variations of crosstalk as a function of coupling length for different wavelengths are shown in Fig. 2. The results show that the crosstalk periodically varies with the coupling length and the local maxima gradually decrease
Fig. 2. Variations of crosstalk as a function of coupling length determined by the theoretical calculations. (Solid, dashed, and dotted lines represent for the operation wavelength of 1.61, 1.565, and 1.53 lm, respectively.)
with the increasing coupling length. In the Ref. [8], the theoretical calculations of the crosstalk and coupling length were chosen at the maximum of the first sidelobe. As the crosstalk of 34 dB at the first maximum is considered at the operating wavelengths of 1.53, 1.565, and 1.61 lm, the coupling lengths of MOC can be correspondingly obtained as 6.10, 6.30, and 6.50 mm. We also observe in Fig. 2 that the coupling length has to be longer than 8.35 mm to get the demanded crosstalk of 35 dB. 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. The analytic theory of MOC 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. In addition, theoretical results do not consider the practical waveguide structure parameters to be fabricated, so these results can only be regarded as a reference. However, due to the condition of coupling waveguide structure for the adiabatic coupler, the effect cannot permanently be ignored. In particular, when the coupling length of the adiabatic coupler is not long enough, such an effect has more impact. Therefore, it is necessary for the numerical analysis to design the coupling waveguide structure of such coupler. In this paper, the waveguide structure parameters are statistically obtained by the BPM of utilizing the full factorial design to satisfy the demanded crosstalk level and required bandwidths at short coupling length. 3. BPM simulation results and discussions Theoretical results do not consider the fabrication of practical waveguide structure parameters, so one can not fabricate a practical MOC based on the theoretical calculations. The purpose of this work is to implement the full factorial design and BPM simulations to obtain the optimal waveguide structure parameters for the MOC with the characteristics of short length, low crosstalk, and C+Lband. The BPM used in this work employs transparent boundary conditions [11,14]. The schematic diagrams of the three-dimensional MOC of VWWS are shown in Fig. 1. Fig. 1a is a plan surface of X–Z, G1 and G2 are the center distances between the two waveguides at z = L/2 and z = 0, W1 and W2 are the widths of guide 1 and guide 2 at z = L, respectively. Fig. 1b shows the cross-section of X–Y, and H is the height of cross-section of waveguide. Before we can proceed to compare the theoretical model and the BPM simulation [11,14], the first essential thing is to transform
C.-F. Chen et al. / Optics Communications 282 (2009) 208–213
the mathematical model to a real device layout. In this case we have assumed that the local beat length be a constant, that is,
CðkÞ ¼
p kb0
we can write
dðkÞ ¼ ;
ð11Þ
jðkÞ ¼
which is fixed through the coupler. But remember the relation in Eq. (2)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðkÞ ¼ d2 ðkÞ þ j2 ðkÞ;
ð12Þ
211
p kb0
p
kb0
cos nðkÞ; sin nðkÞ:
ð13Þ
So the phase constants and the coupling coefficients must obey the two functions given above. The parameters being used in the numerical study of MOC are the waveguide core index nco = 1.543,
Fig. 3. Relations of (a) power evolution vs. propagation distance along z-axis, (b) crosstalk vs. coupling length with polarizations of TE and TM, (c) wavelength band vs. coupling length, (d) crosstalk vs. wavelength, and (e) normalized power vs. propagation distance for the Blackman function weighted sample.
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C.-F. Chen et al. / Optics Communications 282 (2009) 208–213
the waveguide cladding index ncl = 1.528, and minimum local beat length kb0 ¼ 2000 lm at operation wavelength of 1.57 lm. The width parameters W1 and W2 of waveguide structure are selected according to Ref. [15]. The proposed mode calculation of waveguide structures is in the following [15]:
V gW
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2co n2cl ;
ð14Þ
nco where g ¼ operating2pwavelength and W is the width of waveguide structure. The smaller width W2 of the waveguide structure are particularly selected to maintain the single-mode input in the BPM simulations. For a single-mode waveguide, W 6 3.622 lm. For conveniently reveal the results of MOC with the demand of crosstalk in the required bandwidth and coupling length, we define demanded crosstalk (DC) conditions of DC-condition of 35 dB. In additions, in order to use this structural design, the transmission mode of the waveguide has to be single-mode. In this paper, we choose the maximum of the first sidelobe at the curves of crosstalk vs. coupling length. These ranges of coupling lengths still satisfy with the adiabatic condition. We set a crosstalk level and use the full factorial design to acquire the shortest coupling length and conform to the result of C+L-band. By using the full factorial design and BPM simulations, the sample designed structure parameters of MOC to meet the demand of crosstalk and bandwidths within the C+L-band are G1 = 5.2, G2 = 3.4, W1 = 3.4, W2 = 2.6, and H = 3.0 lm. The detail results and discussions are shown in the following sections. In order to understand the effect of mode evolution and switching behavior of MOC, the BPM simulation of mode power along propagation distance of z-axis are shown in Fig. 3a. The mode power launches from left guide 1 and gradually switches into the right guide 2. For the MOC in Fig. 3a, the mode crosstalk must be very small due to the almost flat level of residual power at the output end in guide 1. The data of the power evolutions are recorded with this launch configuration and repeated with different coupling lengths. The crosstalk is defined by 10 times the logarithm of the residual power in guide 1 divided by the input power. According to the conditions of crosstalk, coupling length, and C+L-band, we compare and analyze the simulation results of MOC. To understand the effect of modal polarization, we choose the input wave of MOC as TE and TM modes. Fig. 3b shows the relations of crosstalk vs. coupling length for the TE and TM modes of the input wave. The two curves of TE and TM coincide well and accordingly the polarization sensitivity of MOC is very weak. Furthermore, the structure tolerance can fully support the requested crosstalk of 35 dB and bandwidth of C+L-band in the following results. Fig. 3b indicates that the samples have the coupling length of 3.60 mm at the DC-condition 35 dB by choosing the maximum at the first sidelobe. It was mentioned previously that the variations of crosstalk are related to the operation wavelength. Therefore, to satisfy the demand of C+L-band, the criterion to choose a suitable coupling length for the demand of crosstalk should be made within sufficient ranges of crosstalk. In considering of the bandwidth requirement, we do the BPM simulations and statistically collect the data to obtain the relations between bandwidth and coupling length which is shown in Fig. 3c. In Fig. 3c, the white and black circles represent the lower and upper bound of wavelengths in a specified coupling length, respectively. It can be obtained that the sample has the ranges of coupling length within 3.60–8.00 mm to achieve a bandwidth of within the C+L-band at DC-condition 35 dB. Based on the ranges of coupling length being obtained previously, the simulations are preceded to get the maximum of the first sidelobe coupling length for specified C+L-band. Fig. 3d shows the variations of the crosstalk within the C+L-band for sample with a coupling length of 3.60 mm at DC-condition 35. In DC-condition
35 is considered for the sample, the bandwidths is within 175 nm and the wavelength range is 1.525–1.700 lm. Fig. 3e shows the variations of the powers in the two guides as a function of coupling length along the propagation distance for sample with coupling length of 3.60 mm. In Fig. 3e, we can see the variation of the powers of sample change smoothly from guide 1 to guide 2. The power transfer is stable to ensure the low crosstalk and low attenuation for a given coupling length. In order to provide a practical fabrication, we found separately the best sets of waveguide structure parameters for the MOC with both higher process tolerance and shorter coupling length in the DC-condition of 35 dB. Due to the interacting effect between crosstalk and bandwidth, the data of crosstalk waves are recorded when the crosstalk waveform values are below the demanded crosstalk. Thus, the process tolerance is suitable for the end result calculated by the Blackman weighting function. In fabrication and processing, actually center orientation (G1 and G2) is more accurate than width (W1 and W2). Particularly, the fabricated error of G1 is much smaller. Therefore, the process tolerances of parameters by the full factorial method have the values of G1 = ±0.05, G2 = ±0.2, W1 = ±0.2, and W2 = ±0.2 lm. Under the same conditions, the MOC has a coupling length of 5.0 mm with the parameter H in
Fig. 4. BPM simulations of (a) power evolution varied with propagation distance along z-axis and (b) normalized power of input and output with relative position for MOC.
C.-F. Chen et al. / Optics Communications 282 (2009) 208–213
the ranges of 2.8–3.1 lm. Next, we used process tolerance of boundary values with different operating wavelength to obtained waveguide structures of guarantee single-mode and the curves of the power transfer are smooth. Furthermore, regardless of the input power is large or small in waveguide structures with different operating wavelength, the maintenance of single-mode is guaranteed. The results of similar outcomes are shown in Fig. 3a and e. Next, we adjusted the waveguide structure width at the output end in guide 1, W1, greater than the estimated single-mode structures from the theory. Fig. 4 also illustrates that the mode-interference does not exist in the MOC for propagating a distance of 5.0 mm with the waveguide structure parameters of G1 = 5.1, G2 = 3.4, W1 = 4.5, W2 = 2.6, and H = 3.1 lm. The BPM simulations of (a) power evolution varied with propagation distance along zaxis, and (b) normalized power of input and output with relative position for MOC are shown in Fig. 4. Fig. 4 indicated that the two power curves of input and output coincide well and accordingly the waveform of MOC is very similar. The single-mode is guaranteed with the process tolerances at different operating wavelength. Consequently, structure parameters play an important role in determining the bandwidth under a chosen coupling length and demand of crosstalk. By utilizing the VWWS and optimum structure parameters, the DC-condition 35 is obtained to meet the demand of C+L-band under a coupling length of 3.60 mm. 4. Conclusion The Blackman weighting function is proved to have the superior performances in any aspects of coupling length, bandwidth, cross-
213
talk, and process tolerance. In a proper search of the waveguide parameters with G1 = 5.2, G2 = 3.4, W1 = 3.4, W2 = 2.6 and H = 3.0 lm, which is the sample of MOC, we observe that the curves of the power transfer between the two coupling waveguides are smooth enough to guarantee for broad bandwidth within 175 nm of C+L-band with coupling length of 3.60 mm and crosstalk of 35 dB. The results demonstrate clearly that the selection and design of waveguide structure are very important to obtain the suitable qualities of a MOC. Acknowledgement This work is partially supported by National Science Council, Republic of China, under Contract NSC-97-2221-E-008-019.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of short length and C+L-band mismatched optical coupler with waveguide weighted by the Blackman function Chi-Feng Chen a,*, Yun-Sheng Ku b, Tsu-Te Kung c a
Institute of Opto-Mechatronics Engineering/Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC Department of Mechanical Engineering, National Central University, Jhongli 32054, Taiwan, ROC c Department of Electro-Optical Engineering, National United University, Miaoli 36003, Taiwan, ROC b
a r t i c l e
i n f o
Article history: Received 26 March 2008 Received in revised form 1 October 2008 Accepted 1 October 2008
Keywords: Optical coupler Adiabatic directional couplers Mismatched optical coupler Coupling waveguide structure Blackman weighting function
a b s t r a c t A mismatched optical coupler with waveguide weighted by the Blackman function is numerically investigated in the demand of short length, C+L-band, and low crosstalk. Utilizing the full factorial design, the structure parameters of coupling waveguide are obtained by beam propagation method. In the condition of crosstalk of 35 dB, the mismatched optical coupler with proper selected waveguide structure parameters is found to have a coupling length of 3.60 mm in the transmission wavelength ranges of C+L-band (1.53–1.61 lm). Obviously, the selection and design of waveguide structure are very important to satisfy the qualities of a mismatched optical coupler for the demand of short length, broad bandwidth, and low crosstalk. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Optical couplers are the 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 with switching capability, the mismatched optical coupler (MOC) is a suitable choice to have the features of transferring power from one guide to the other and establishing an adiabatic coupling in the core layer. Previously, the MOC was proposed and theoretically analyzed in the microwave applications [1–3]. Due to the microwave ranges between millimeter and meter, it seems impractical for such coupler having the length of 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 which has 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 * Corresponding author. Tel.: +886 3 4267308; fax: +886 3 4254501. E-mail address: [email protected] (C.-F. Chen). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.012
efficiency which applied only to full couplers with long path lengths. Several weighting functions were applied to the MOC to proceed with the simulation and fabrication [7–9]. The variable width of waveguide structures (VWWS) of MOC were constrained by these weighting functions. The results demonstrate that the efficiency for the VWWS is obviously influenced by the different weighting functions. Among the weighting functions, the Blackman function was proved to have the preferable performances for a MOC. Significant improvement in crosstalk level of 33.67 dB and coupling length of 8.2 mm were also obtained in a tapered velocity coupler weighted by Blackman function [7]. The MOC with the variable width of waveguide structure was proposed and optimally designed [8,9]. Utilizing the sin-square and raised-cosine weighting functions on the MOC, the coupling length/bandwidths were obtained to be 6.25 mm/290 nm and 5.8 mm/220 nm, respectively under the crosstalk of 35 dB. In addition, the criteria of adiabatic power flow along tapered mismatched directional couplers were presented [10]. 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 [11]. In this paper, we adopt the coupled mode equations established previously for the tapered coupler. After then, we use the Fourier
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transform to express the dynamic power flow in the domain of normalized propagation distance in order for the Blackman weighting function to be applied and integrated to obtain the crosstalk. We investigate the MOC with VWWS weighted by the Blackman function. The optimum waveguide structure parameters are obtained by the beam propagation method (BPM) [8,9,11] of utilizing the full factorial method [12]. The performances of coupler length, crosstalk, process tolerances, and C+L-band for the MOC are studied in detail through the discussions. It is shown that the waveguide structure parameters are crucial in determining the required optical properties of MOC. 2. Theoretical model The MOC consists of two waveguides as shown in Fig. 1. The power launches into guide 1 at z = 0 and travels along in the z direction. We focus on the forward normal modes in the co-directional coupler and ignore the backward normal modes. The following equations govern the two forward normal modes for such a waveguide system. The wave amplitudes for the two coupled waveguides can be written in the following:
dA1 ðzÞ ¼ j½b1 ðzÞ þ jðzÞA1 ðzÞ þ jjðzÞA2 ðzÞ; dz dA2 ðzÞ ¼ jjðzÞA1 ðzÞ j½b2 ðzÞ þ jðzÞA2 ðzÞ; dz
ð1Þ
where A1,2(z) are wave amplitudes in guide 1 and 2, respectively, b1,2(z) are uncoupled local-phase constants of guides 1 and 2, respectively. The mutual and self-coupling coefficients j(z) are as-
sumed to be identical. j(z) also must be a real value to satisfy the conservation of energy. The wave amplitudes are normalized so that the power in either guide is equal to the square of the wave amplitudes. The phase constants of the two composite guides vary in the reverse direction and cross over at some points of the coupler. Several symbols being used in the tapered power equations are defined as [3]
bðzÞ ¼ 1=2½b1 ðzÞ þ b2 ðzÞ; dðzÞ ¼ 1=2½b2 ðzÞ b1 ðzÞ; dðzÞ cot nðzÞ ¼ ; jðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p CðzÞ ¼ ¼ d2 ðzÞ þ j2 ðzÞ ¼ jðzÞ 1 þ M2 ; kb0 ðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðzÞ CðzÞ þ dðzÞ cos ¼ ; 2 2CðzÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s nðzÞ CðzÞ dðzÞ ; sin 2 2CðzÞ
ð2Þ
where M is termed the mismatch value that measures the synchronicity of the guides, cot n(z) is the ratio of variation of phase constant to variation of coupling coefficient, and kb0 ðzÞ is the minimum local beat length. For symmetry, b = constant and d(0) P 0. It is considered that d(z) and j(z) vary slowly compared to kb0 ðzÞ. And the traveling wave in the opposite direction is neglected. The characteristic impedances are normalized so that the power in either guide is equal to the square of the wave amplitude. The powers of single tapered mode directional couplers in the two guides are expressed as [3]
P1 ðzÞ ¼ jA1 ðzÞj2 ¼ cos2
nðzÞ 2 nðzÞ jN A ðzÞj2 sin jNB ðzÞj2 2 2
sinðnðzÞÞReðNA ðzÞNB ðzÞÞ; nðzÞ 2 nðzÞ jN A ðzÞj2 cos2 jNB ðzÞj2 P2 ðzÞ ¼ jA2 ðzÞj2 ¼ sin 2 2
ð3Þ
sinðnðzÞÞReðNA ðzÞNB ðzÞÞ; where A1,2(z) are the wave amplitudes in guide 1 and 2, and NA,B(z) are the in-phase local-normal and out-phase local-normal modes, respectively. The function NB ðzÞ represents the complex conjugate of NB(z) and ReðN A ðzÞN B ðzÞÞ indicates the real part of N A ðzÞN B ðzÞ. And NA(z) and NB(z) are approximately given by [3]
Z z 1 dn 2jqðz0 Þ 0 NA ðzÞ ffi ejqðzÞ N A ð0Þ þ N B ð0Þ dz 0e 2 0 dz ! Z 0 Z z 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 NA ð0Þ dz dz ; 0e 00 e 4 0 dz 0 dz Z z 1 dn 2jqðz0 Þ 0 NB ðzÞ ffi ejqðzÞ NB ð0Þ N A ð0Þ dz 0e 2 0 dz ! Z z Z 0 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 dz dz ; NB ð0Þ 0e 00 e 4 dz 0 0 dz
Fig. 1. Schematics diagrams of the MOC with three-dimensional variable widths of waveguide structure viewed in the (a) plan surface of X–Z and (b) cross-section of X–Y.
ð4Þ
Rz 0 where qðzÞ ¼ 0 Cðz0 Þdz jN A ð0Þj2 and jNB ð0Þj2 represent the amounts of power excited in the in-phase local-normal mode and out-phase local-normal mode, respectively. The amplitudes are normalized so that jNA ð0Þj2 þ jN B ð0Þj2 ¼ 1. Let n(z) = 0 at z = 0 in Eq. (1), the powers at the input ends can be expressed as P1 ð0Þ ¼ jN A ð0Þj2 and P 2 ð0Þ ¼ jN B ð0Þj2 , respectively. Assume the power initially exists at the input end in guide 1 such that P1(0) = 1 and P2(0) = 0. Thus only one of the local-normal modes will be excited under such conditions, then NA(0) = 1 and NB(0) = 0. The dynamic power flows in the two guides can be written as
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1 2 1 P1 ðzÞ ffi cos2 nðzÞ f1 þ mðzÞg þ sin nðzÞ lðzÞ þ sinðnðzÞÞDðzÞ; 2 2 2 1 2 1 P2 ðzÞ ffi sin nðzÞ f1 þ mðzÞg þ cos nðzÞÞlðzÞ sinðnðzÞ DðzÞ; 2 2 ð5Þ where
2 Z 1 z dn 2jqðz0 Þ 0 lðzÞ ¼ e dz ; 4 0 dz0 ! Z 0 Z z 1 dn 2jqðz0 Þ z dn 2jqðz00 Þ 00 0 mðzÞ ¼ Re dz dz ¼ lðzÞ; 0e 00 e 2 dz 0 0 dz Z z 1 dn 2jqðz0 Þ 0 DðzÞ ¼ Re e2jqðzÞ dz : 0e 2 0 dz
ð6Þ
Let l(z) + v(z) = 0 to satisfy the energy conservation and n(l) = p for a coupler substitute into Eqs. (1) and (4) to obtain the powers in the following:
P1 ðlÞ ¼ jNB ðlÞj2 ¼ lðlÞ; P2 ðlÞ ¼ jNA ðlÞj2 ¼ 1 lðlÞ:
ð7Þ
In Eq. (6), l(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 local-normal mode, this kind of coupler is known as an adiabatic directional coupler. In addition, we can express the mode crosstalk lðlÞ in Fourier transform as [13]
Z 1 l lðlÞ ¼ 4 0 Z 1 l ¼ 4 0
2 2 Z R0 dn 2jqðz0 Þ 0 1 l dn j2l s 00 0 00 0 dz ¼ Cðs Þds ðlds Þ 0e 0 e 4 0 lds dz 2 Z 2 dn jul 0 1 g dn iul e du ; 0 e ds ¼ 4 0 du ds
ð8Þ
R s0 00 where u ¼ 2 0 Cðs00 Þds ; g ¼ uðlÞ. The crosstalk related to the coupling length l is proportional to the square of the Fourier transform dn .In this study, we are interested in the constant local beat of du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wavelength of coupler, therefore CðzÞ ¼ d2 ðzÞ þ j2 ðzÞ ¼ kp = conb0 stant. The crosstalk formula of Eq. (8) can be rewritten as
2 Z 2 dn 2jqðz0 Þ 0 1 k dn jk2pl 0 0 b0 k dk e dz ¼ e 4 0 dz0 4 0 dk0 2 Z 0 1 k 0 0 ¼ f ðk Þej2pDk dk ; 4 0 Z
1 lðk; DÞ ¼
z
ð9Þ
0
dn where tapered function f ðk Þ ¼ dk 0 , normalized propagation distance k ¼ zl, and normalized coupler length D ¼ k l . Due to the coupling b0 coefficient equal to zero at normalized propagation distances z/l = 0 and z/l = 1, there will exist infinite gap parameter G2 at these points [10]. In order to obtain a practical device, we introduce the truncation range of 0.05 6 z/l 6 0.95 to do the simulations [7]. One can see that it resembles the Fourier transform with k = 0.95 and f(k) is zero outside the range of 0.05 6 k 6 0.95. On the other hand, the crosstalk at the ends of guides l(1, D) represents the frequency domain of f(k). To study the crosstalk of MOC, we select the Blackman weighting function [7] f(k) written in the following:
f ðkÞ ¼ 1 1:19 cosð2pkÞ þ 0:19 cosð4pkÞ:
ð10Þ
Eqs. (9) and (10) are generally used to study the crosstalk of MOC. To understand the effect of operation wavelength in this coupling, the theoretical calculations of the variations of crosstalk as a function of coupling length for different wavelengths are shown in Fig. 2. The results show that the crosstalk periodically varies with the coupling length and the local maxima gradually decrease
Fig. 2. Variations of crosstalk as a function of coupling length determined by the theoretical calculations. (Solid, dashed, and dotted lines represent for the operation wavelength of 1.61, 1.565, and 1.53 lm, respectively.)
with the increasing coupling length. In the Ref. [8], the theoretical calculations of the crosstalk and coupling length were chosen at the maximum of the first sidelobe. As the crosstalk of 34 dB at the first maximum is considered at the operating wavelengths of 1.53, 1.565, and 1.61 lm, the coupling lengths of MOC can be correspondingly obtained as 6.10, 6.30, and 6.50 mm. We also observe in Fig. 2 that the coupling length has to be longer than 8.35 mm to get the demanded crosstalk of 35 dB. 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. The analytic theory of MOC 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. In addition, theoretical results do not consider the practical waveguide structure parameters to be fabricated, so these results can only be regarded as a reference. However, due to the condition of coupling waveguide structure for the adiabatic coupler, the effect cannot permanently be ignored. In particular, when the coupling length of the adiabatic coupler is not long enough, such an effect has more impact. Therefore, it is necessary for the numerical analysis to design the coupling waveguide structure of such coupler. In this paper, the waveguide structure parameters are statistically obtained by the BPM of utilizing the full factorial design to satisfy the demanded crosstalk level and required bandwidths at short coupling length. 3. BPM simulation results and discussions Theoretical results do not consider the fabrication of practical waveguide structure parameters, so one can not fabricate a practical MOC based on the theoretical calculations. The purpose of this work is to implement the full factorial design and BPM simulations to obtain the optimal waveguide structure parameters for the MOC with the characteristics of short length, low crosstalk, and C+Lband. The BPM used in this work employs transparent boundary conditions [11,14]. The schematic diagrams of the three-dimensional MOC of VWWS are shown in Fig. 1. Fig. 1a is a plan surface of X–Z, G1 and G2 are the center distances between the two waveguides at z = L/2 and z = 0, W1 and W2 are the widths of guide 1 and guide 2 at z = L, respectively. Fig. 1b shows the cross-section of X–Y, and H is the height of cross-section of waveguide. Before we can proceed to compare the theoretical model and the BPM simulation [11,14], the first essential thing is to transform
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the mathematical model to a real device layout. In this case we have assumed that the local beat length be a constant, that is,
CðkÞ ¼
p kb0
we can write
dðkÞ ¼ ;
ð11Þ
jðkÞ ¼
which is fixed through the coupler. But remember the relation in Eq. (2)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CðkÞ ¼ d2 ðkÞ þ j2 ðkÞ;
ð12Þ
211
p kb0
p
kb0
cos nðkÞ; sin nðkÞ:
ð13Þ
So the phase constants and the coupling coefficients must obey the two functions given above. The parameters being used in the numerical study of MOC are the waveguide core index nco = 1.543,
Fig. 3. Relations of (a) power evolution vs. propagation distance along z-axis, (b) crosstalk vs. coupling length with polarizations of TE and TM, (c) wavelength band vs. coupling length, (d) crosstalk vs. wavelength, and (e) normalized power vs. propagation distance for the Blackman function weighted sample.
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the waveguide cladding index ncl = 1.528, and minimum local beat length kb0 ¼ 2000 lm at operation wavelength of 1.57 lm. The width parameters W1 and W2 of waveguide structure are selected according to Ref. [15]. The proposed mode calculation of waveguide structures is in the following [15]:
V gW
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2co n2cl ;
ð14Þ
nco where g ¼ operating2pwavelength and W is the width of waveguide structure. The smaller width W2 of the waveguide structure are particularly selected to maintain the single-mode input in the BPM simulations. For a single-mode waveguide, W 6 3.622 lm. For conveniently reveal the results of MOC with the demand of crosstalk in the required bandwidth and coupling length, we define demanded crosstalk (DC) conditions of DC-condition of 35 dB. In additions, in order to use this structural design, the transmission mode of the waveguide has to be single-mode. In this paper, we choose the maximum of the first sidelobe at the curves of crosstalk vs. coupling length. These ranges of coupling lengths still satisfy with the adiabatic condition. We set a crosstalk level and use the full factorial design to acquire the shortest coupling length and conform to the result of C+L-band. By using the full factorial design and BPM simulations, the sample designed structure parameters of MOC to meet the demand of crosstalk and bandwidths within the C+L-band are G1 = 5.2, G2 = 3.4, W1 = 3.4, W2 = 2.6, and H = 3.0 lm. The detail results and discussions are shown in the following sections. In order to understand the effect of mode evolution and switching behavior of MOC, the BPM simulation of mode power along propagation distance of z-axis are shown in Fig. 3a. The mode power launches from left guide 1 and gradually switches into the right guide 2. For the MOC in Fig. 3a, the mode crosstalk must be very small due to the almost flat level of residual power at the output end in guide 1. The data of the power evolutions are recorded with this launch configuration and repeated with different coupling lengths. The crosstalk is defined by 10 times the logarithm of the residual power in guide 1 divided by the input power. According to the conditions of crosstalk, coupling length, and C+L-band, we compare and analyze the simulation results of MOC. To understand the effect of modal polarization, we choose the input wave of MOC as TE and TM modes. Fig. 3b shows the relations of crosstalk vs. coupling length for the TE and TM modes of the input wave. The two curves of TE and TM coincide well and accordingly the polarization sensitivity of MOC is very weak. Furthermore, the structure tolerance can fully support the requested crosstalk of 35 dB and bandwidth of C+L-band in the following results. Fig. 3b indicates that the samples have the coupling length of 3.60 mm at the DC-condition 35 dB by choosing the maximum at the first sidelobe. It was mentioned previously that the variations of crosstalk are related to the operation wavelength. Therefore, to satisfy the demand of C+L-band, the criterion to choose a suitable coupling length for the demand of crosstalk should be made within sufficient ranges of crosstalk. In considering of the bandwidth requirement, we do the BPM simulations and statistically collect the data to obtain the relations between bandwidth and coupling length which is shown in Fig. 3c. In Fig. 3c, the white and black circles represent the lower and upper bound of wavelengths in a specified coupling length, respectively. It can be obtained that the sample has the ranges of coupling length within 3.60–8.00 mm to achieve a bandwidth of within the C+L-band at DC-condition 35 dB. Based on the ranges of coupling length being obtained previously, the simulations are preceded to get the maximum of the first sidelobe coupling length for specified C+L-band. Fig. 3d shows the variations of the crosstalk within the C+L-band for sample with a coupling length of 3.60 mm at DC-condition 35. In DC-condition
35 is considered for the sample, the bandwidths is within 175 nm and the wavelength range is 1.525–1.700 lm. Fig. 3e shows the variations of the powers in the two guides as a function of coupling length along the propagation distance for sample with coupling length of 3.60 mm. In Fig. 3e, we can see the variation of the powers of sample change smoothly from guide 1 to guide 2. The power transfer is stable to ensure the low crosstalk and low attenuation for a given coupling length. In order to provide a practical fabrication, we found separately the best sets of waveguide structure parameters for the MOC with both higher process tolerance and shorter coupling length in the DC-condition of 35 dB. Due to the interacting effect between crosstalk and bandwidth, the data of crosstalk waves are recorded when the crosstalk waveform values are below the demanded crosstalk. Thus, the process tolerance is suitable for the end result calculated by the Blackman weighting function. In fabrication and processing, actually center orientation (G1 and G2) is more accurate than width (W1 and W2). Particularly, the fabricated error of G1 is much smaller. Therefore, the process tolerances of parameters by the full factorial method have the values of G1 = ±0.05, G2 = ±0.2, W1 = ±0.2, and W2 = ±0.2 lm. Under the same conditions, the MOC has a coupling length of 5.0 mm with the parameter H in
Fig. 4. BPM simulations of (a) power evolution varied with propagation distance along z-axis and (b) normalized power of input and output with relative position for MOC.
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the ranges of 2.8–3.1 lm. Next, we used process tolerance of boundary values with different operating wavelength to obtained waveguide structures of guarantee single-mode and the curves of the power transfer are smooth. Furthermore, regardless of the input power is large or small in waveguide structures with different operating wavelength, the maintenance of single-mode is guaranteed. The results of similar outcomes are shown in Fig. 3a and e. Next, we adjusted the waveguide structure width at the output end in guide 1, W1, greater than the estimated single-mode structures from the theory. Fig. 4 also illustrates that the mode-interference does not exist in the MOC for propagating a distance of 5.0 mm with the waveguide structure parameters of G1 = 5.1, G2 = 3.4, W1 = 4.5, W2 = 2.6, and H = 3.1 lm. The BPM simulations of (a) power evolution varied with propagation distance along zaxis, and (b) normalized power of input and output with relative position for MOC are shown in Fig. 4. Fig. 4 indicated that the two power curves of input and output coincide well and accordingly the waveform of MOC is very similar. The single-mode is guaranteed with the process tolerances at different operating wavelength. Consequently, structure parameters play an important role in determining the bandwidth under a chosen coupling length and demand of crosstalk. By utilizing the VWWS and optimum structure parameters, the DC-condition 35 is obtained to meet the demand of C+L-band under a coupling length of 3.60 mm. 4. Conclusion The Blackman weighting function is proved to have the superior performances in any aspects of coupling length, bandwidth, cross-
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talk, and process tolerance. In a proper search of the waveguide parameters with G1 = 5.2, G2 = 3.4, W1 = 3.4, W2 = 2.6 and H = 3.0 lm, which is the sample of MOC, we observe that the curves of the power transfer between the two coupling waveguides are smooth enough to guarantee for broad bandwidth within 175 nm of C+L-band with coupling length of 3.60 mm and crosstalk of 35 dB. The results demonstrate clearly that the selection and design of waveguide structure are very important to obtain the suitable qualities of a MOC. Acknowledgement This work is partially supported by National Science Council, Republic of China, under Contract NSC-97-2221-E-008-019.
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