Spectral shift by half free-spectral-range for microring resonator employing the phase jump phenomenon in coupled-waveguide and application on all-microring wavelength interleaver

Using coupled-mode theory, we have shown that there is a π phase jump between the input and the through/drop fields of a codirectional coupler when the gap width between the coupled-waveguides reaches certain values such that the length of the coupler equals to the odd integer (for through field) or even integer (for drop field) times of the Transfer Distance. We introduced an efficient numerical method based on combining the scattering matrix method and FDTD method for analyzing a microring that has material loss. By applying this method, we found that the phase jump phenomenon also occurs in a half-ring coupler when the gap width between the coupled half-ring waveguides reaches a critical value. We showed that, for a given operating bandwidth, it is important that the gap width between the rings has to be larger than a certain value in order to avoid the phase jump, or smaller in order to take advantage of the phase jump. Based on the phase jump phenomenon, we found that the through and the drop spectra of the single-arm and the double-arm microring can be manipulated to shift about one half free spectral range by selecting appropriate gap widths. A novel all-microring wavelength interleaver, based on the phase jump phenomenon, is proposed and numerically demonstrated. © 2009 Optical Society of America OCIS codes: (130.1750) Components; (130.2790) Guided waves; (130.3120) Integrated optics devices; (130.7408) Wavelength filtering devices; (230.3120) Integrated optics devices; (230.4555) Coupled resonators; (230.5750) Resonators; (230.7380) Waveguides, channeled. References and links 1. M. C. M. Lee and M. C. Wu, “Tunable coupling regimes of silicon microdisk resonators using MEMS actuators,” Opt. Express 14, 4703–4712 (2006). 2. A. Yariv and P. Yeh, Photonics:optical electronics in modern communications (Oxford University Press Inc., 2007), pp. 184-189. 3. M. A. Popovic, C. Manolatou, and M. R. Watts, “Coupled-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express 14, 1208–1222 (2006). 4. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Effect of a layered environment on the complex nature frequencies of two-dimensional WGM dieletric-ring resonators,” IEEE J. Lightwave Technol. 20, 1563– 1572 (2002). #107951 $15.00 USD Received 25 Feb 2009; revised 3 Apr 2009; accepted 19 Apr 2009; published 27 Apr 2009 (C) 2009 OSA 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 7756 5. O. Schwelb, “On the nature of resonance splitting in coupled multiring optical resonators,” Opt. Commun. 281, 1065–1071 (2008). 6. T. Barwicz, M. A. Popovic, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microringresonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12, 1437–1442 (2004). 7. T. Barwicz, M. A. Popovic, M. R. Watts, P. T. Rakich, E. P. Ippen, and H. I. Smith, “Fabrication of add-drop filters based on frequency-matched microring resonators,” IEEE J. Lightwave Technol. 24, 2207–2218 (2006). 8. S. Cao, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, S. Suzuki, K. Y. Wu, and P. Xie, “Interleaver technology: comparisons and applications requirements,” IEEE J. Lightwave Technol. 22, 281–289 (2004). 9. C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (John Willey & Sons Inc., 1999), pp. 165-177. 10. T. Mizuno, T. Kitoh, M. Oguma, Y. Inoue, T. Shibata and H. Takahashi, “Uniform wavelength spacing MachZehnder interference using phase-generating couplers,” IEEE J. Lightwave Technol. 24, 3217–3226 (2006). 11. K. Oda, N. Takato, H. Toba, and K. Nosu, “A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems,” IEEE J. Lightwave Technol. 6, 1016–1023 (1988). 12. M. Kohtoku, S. Oku, Y. Kadota, and Y. Yoshikuni, “200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach-Zehnder interference with a ring resonator,” IEEE Photon. Technol. Lett. 12, 1174–1176 (2000). 13. Z. Wang, S. J. Chang, C. Y. Ni, and Y. J. Chen, “A high-performance ultracompact optical interleaver based on double-ring assisted Mach-Zehnder interferometer,” IEEE Photon. Technol. Lett. 19, 1072–1074 (2007). 14. J. Song, Q. Fang, S. H. Tao, M. B. Yu, G. Q. Lo, and D. L. Kwong, “Passive ring-assisted Mach-Zehnder interleaver on silicon-on-insulator,” Opt. Express 16, 8359–8365 (2008). 15. C. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000), Chap. 4. 16. M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kartner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. 31, 2571–2573 (2006). 17. S. Xiao, M. H. Khan, H. Shen, and M. Qi, “A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion,” Opt. Express 15, 14765–14771 (2007). 18. S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Silicon-on-insulator microring add-drop filters with free spectral ranges over 30 nm,” IEEE J. Lightwave Technol. 26, 228–236 (2008). 19. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).


Introduction
Changing the gap width between the coupled waveguides is desired in situations when the power coupling coefficient needs to be tuned for purpose such as changing the power coupling efficiency, obtaining the critical coupling [1,2].There is a π/2 phase change for weak coupling and this phase change departs from π/2 as the coupling conditions deviates from weak coupling causing the coupling-induced resonance frequency shifts (CIFS) for a resonator [3][4][5].The magnitude of the phase deviation from π/2 in the CIFS effect is small but for telecomgrade devices where precise wavelength control is required, the CIFS effect has to be avoided or compensated [6,7].In this paper, we report and analyze our findings on a large phase change, a π phase jump, for the coupled waveguides when the gap width between the coupled waveguides changes.We demonstrated and analyzed in Sec.2.1 that the phenomenon occurs in the conventional straight co-directional coupler, and we showed in Sec.2.2 by numerical example that the phenomenon also occurs in half-ring coupler.Basic characteristics and discussions of the phenomenon with respect to the gap width variation were given.Spectral shift, based on the phase jump phenomenon, by half free spectral range (FSR) in a single-arm and a double-arm microring resonator were demonstrated in Sec.2.3.
For processing the dense channel spacing signal in optical communication, it is sometimes desired that the signals be interleaved such that the communication bandwidth or capacities can be increased in a cost-effective way.There are three approaches to achieve dense signals interleaving: lattice filter, Gires-Torunois-based Michelson interferometer and array-waveguide grating [8].Bulk optics and integrated waveguide optics can be implemented for the interleaver such that a series of dense signal channels can be combined or separated.Waveguide-type interleaver has the advantage of compact size, moreover, compact waveguide-type interleaver with multi-stage design is more flexible than bulk one with multi-stage design.For the waveguidetype interleaver, Mach-Zehnder (MZ) interleaver [9,10] was commonly used, and multi-stage MZ interleaver and microring-assisted MZ interleaver [11][12][13][14] were used to widen the 3 dB bandwidth and reduce the crosstalk.However, these devices still suffer from the large footprint, usually several millimeters to centimeters, to make the integration impractical.In this paper, we propose an all-microring interleaver based on employing the phase jump phenomenon, and chip size integration could be implemented for the all-microring device.A numerical example is given to demonstrate the feasibility of the idea in Sec. 3.

Symmetrical co-directional coupler
Figure 1 shows a symmetrical co-directional coupler.The input-output relationship derived Fig. 1.Symmetrical co-directional coupler from the rigorous coupled-mode theory (rCMT), as given by Ref. 15, can be recast into a simple form in terms of the scattering matrix or S-matrix: where where L is the coupling length, K a = (K −CX)/(1 − |C| 2 ), and Γ = (KC − X)/(1 − |C| 2 ).K, C, and X are defined as: E 1 and E 2 are the mode fields of the waveguide 1 and 2, respectively.L c = π/(2K a ) is defined as the Transfer Distance.2L c is the coupling length for one complete cycle of power exchange.K is the coupling coefficient, C represents the cross-coupling coefficient, and X is the self-coupling coefficient of the coupled modes for the symmetrical co-directional coupler.When B 0 =0, the input-output relationship is: For weak coupling (wCMT), i.e.C and X are equal to zero, Γ=0, K a =K, and L c =π/(2K), the matrix elements are given as: When the gap width g is varied, K, C, X and hence K a , Γ and L c are varied accordingly, and up to a certain gap width the cosine and the sine function in Eq. (2a)-(2d) change sign, a π phase jump occur in s 11 , s 21 , s 12 and s 22 .From the definition of L c and Eq.(2a)-(2d), the phase jump occurs at the gap width for which is satisfied, n= odd integer for s 11 and s 22 , n= even integer for s 12 and s 21 .The underlying physics of the π phase jump is fundamentally due to the phase accumulation of the back and the forth coupling each contributes π/2 phase, n + 1 is the number of the forth and the back coupling within L. Notice that for B 0 =0, the phase difference between A 0 and A 1 is φ 11 , and the phase difference between A 0 and B 1 is φ 21 , according to Eq. ( 6) and (7).
We demonstrate this phenomenon with a numerical example as follows: A co-directional coupler as shown in Fig. 1 is given with n core =2.5, n cladding =1.5, a=300 nm, L= 6 um, and λ =1550 nm.The single TE mode field distribution and the effective mode index for the single waveguide were obtained by using the beam propagation method (BPM) and they were then substituted into Eq.( 3), ( 4), (5) and Eq. ( 2a)-(2d) to obtain all the matrix elements in Eq. (1b) and L c for the co-directional coupler.Both rigorous coupled-mode theory (rCMT) and weak coupling assumption (wCMT) were calculated.The results for K a L/π, L c and the magnitude and phase for s 11 and s 21 vs. the gap width are shown in Figs.2(a)-2(f).s 22 and s 12 are not shown but they are related to s 11 and s 21 by: s 22 = s 11 , s 12 = s 21 as can be seen from Eq (2a)-(2d).
Figures 2(a)-2(f) reveal, firstly, L c decreases with the gap width as expected, secondly, the π phase jump occurred for wCMT as well as for rCMT as predicted, there is a phase difference of ΓL between wCMT and rCMT consistent with Eq.( 2a)-(2d), thirdly, the magnitude of the matrix element has a minimum at the gap width where there is phase jump, and fourthly, only limited number of phase jump occurred as the gap width is varying; the odd order phase jump occurs for φ 11 and the even order phase jump occurs for φ 21 .According to Eq. ( 5) mathematically, the phase jump will keep on occurring with n >4 but at negative gap width that is not realistic and the CMT would not be valid, therefore, only limited number of phase jump occurs for a given coupling length L. When L increases, the first-order jump (n=1) occurs at larger gap width and more higher order jumps occur at smaller gap width.

Symmetrical half-ring coupler
In order to analyze the phase jump phenomenon for a half-ring coupler as shown in Fig. 3, we have developed an effective numerical method for numerically measuring the matrix elements of the S-matrix.Similar method was given by Ref. 3 for straight-half ring coupler.Our method is introduced as follows: Step1: the BPM is used to obtain the eigen-mode field distribution (M1) of a straight waveguide that has a cross-section identical to that of the ring waveguide.
Step 2: M1 is taken as the launch field for a quarter ring waveguide in the FDTD simulation, the output filed of the quarter ring is then used as the input field for the second iteration, and the output field converged to the eigen-mode, M2, of the ring waveguide after several iterations.
Step 3: M2 is then taken as the launch field at plane 1, referring to Fig. 3, with a specific pulse form and duration in the FDTD simulation, and the time evolution of the field at plane 1, 2 and 4 are monitored.
Step 4: discrete Fourier transform is carried out on the time-domain fields to obtain the spectra for A 0 , A 1 and B 1 .The matrix elements of S-matrix, s 11 and s 21 , are then obtained through Eq. ( 6) and (7).
Step 3 and 4 are repeated with launching M2 at plane 3, s 22 and s 12 are obtained accordingly.A numerical example, referring to Fig. 3, was carried out with the following parameters: n core =2.5, n cladding =1.5, waveguide width=300 nm, λ =1550 There is a noticeable phase protruding about 0.1π in φ 11 right below the gap width of the phase jump, no satisfactory explanation could be given for the protrude phase.Nevertheless, it is clear that the phase jump indeed occurs in the half-ring coupler and its magnitude is close to π.The phase jump occurred at the minimum s 11 and maximum s 21 .There is only one phase jump for φ 11 and no phase jump for φ 21 in this case.The criteria for the phase jump to occur in the half-ring coupler could be modified from Eq. ( 10) as L e f f =n L c,e f f , where L e f f is the effective coupling length and L c,e f f is the effective Transfer Distance, and n=1 in this particular example.For half-ring coupler with larger radius of curvature with the same gap width, we expect that L e f f would be larger and L c,e f f would be smaller such that the number of phase jump would be increased as the gap width is varying.
When the gap width and the radius of curvature for the half-ring coupler are fixed, the phase jump phenomenon will occur when the wavelength is varying.The simulation results are shown in Figs.5(a) and 5(b) for the same half-ring coupler but fixed gap width at 85 nm.The phase jump occurs at 1481.14 nm for this case.We define this wavelength as λ c .For longer wavelength, the evanescent field extends further out, a shorter L c,e f f is expected, and more phase jump would occur at longer wavelengths.The coefficient of K in Eq. ( 3) is inversely proportional to λ , the cosine and the sine function in Eq. ( 2a)-(2d) change sign whenever the  wavelength changes by the order of magnitude of L, we therefore assert that the wavelength difference between two adjacent phase jump of the same order would be in the order of L e f f for half-ring coupler.
We have calculated λ c vs. gap width, g, and the result is shown in Fig. 6.Several important aspects with respect to the phase jump phenomenon, when the device is operated within a certain bandwidth, are implied: firstly, if one wants to avoid the phase jump, the gap width has to be larger than a certain value determined by the upper bound of the operating bandwidth, in the region of g III as shown in Fig. 6 for C-band operation as an example, secondly, if one wants to utilize the phase jump for all operating wavelength, the gap width has to be smaller than a certain value determined by the lower bound of the operating bandwidth, in the region of g I , and thirdly, if one wants to utilize the phase jump within the operating wavelength range, then the gap width has to be controlled and varied within a certain range g II determined by the upper and the lower bound of the operating bandwidth.

Single-arm and double-arm microring
Referring to Fig. 4(b), we designated the gap width as g C at the phase jump, g L when there is no phase jump, and g S when there is phase jump.For a single-arm microring with g S gap width, the through field undergoes π phase jump and the field that is coupled back to the bus after one round trip in the ring undergoes twice π/2 phase change from the coupling, the over-all through field at resonance is in constructive interference in contrast to that of the microring with g L gap width where the over-all through field at resonance is in destructive interference, therefore, we expect that there will be a spectral shift between the microrings with g L and g S gap width, the amount of the spectral shift will be around half the free spectral range (FSR).For a double-arm microring, we define that the microring with both gap width is g L or g S as type I, and the microring with one gap is g L and the other gap is g S as type II.For type I, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) same as the through field such that the over-all through field at resonance is in destructive interference due to twice π/2 phase change from the coupling, therefore, we expect that there is no spectral shift, neither in the through port nor in the drop port for type I, with respect to the single-arm microring with g L .For type II, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) that is opposite to the through field such that the over-all through field at resonance is in constructive interference, therefore, we expect that there is a half FSR spectral shift both in the through port and in the drop port for type II.
We prove the above-mentioned assertions by the following numerical examples.The microring is composed of half-ring bus and a ring with identical parameters, i.e β matching.The spectrum at the through port I through and the spectrum at the drop port I drop were calculated by the analytical expressions, Eq. (A.5), Eq. (A.14), and Eq.(A.15), as given in Appendix A by using the elements of the S-matrix for the half-ring coupler.Numerical measurement on the S-matrix elements of the half-ring coupler as introduced in Sec.2.2 was performed first, and then the measured matrix elements were substituted into the analytical expressions in Appendix A to obtain the I through and the I drop spectra.We have also applied the FDTD method to obtain I through and I drop spectra in order to verify the validity of the analytical expressions.Same parameters for the half-ring coupler in Sec.2.2 were used here except that the material power loss coefficient was chosen to be 50 dB/cm for the microring in order to, firstly, reduce the computational time for the FDTD method and secondly, profound spectra that is close to critical coupling could be obtained.
Figure 7(a) and 7(b) show the results for single-arm microring obtained by the FDTD method, red solid lines, and by the analytical expressions with the S-matrix elements, black solid lines.Both methods gave identical results except that the dip depths are slightly different due to the limited FDTD computational time.Fig. 7(a) shows I through spectrum for a single-arm microring with g L =350 nm, i.e. no phase jump.Fig. 7(b) shows I through spectrum for the singlearm microring with g S =20 nm, i.e. with phase jump.It is apparent that the spectra are shifted with each other by approximately half the FSR.
The reason that the shift is not exactly half FSR is that the resonance frequency is slightly different between these two cases due to that φ 21 is not exactly π/2 for g S =20 nm; weak coupling assumption is not exactly applied to g S =20 nm.This assertion can be confirmed from Fig. 4(b) and from the non-zero ΓL in Fig. 2(c) for the co-directional coupler.This is basically the coupling-induced resonance frequency shift (CIFS) phenomenon when weak coupling assumption is not exactly applied [3].
Figure 8(a) and 8(b) show the through and the drop spectra of the type I and the type II double-arm microrings, respectively, with g L =350 nm and g S =20 nm.Again, the resonance frequencies are slightly different due to the CIFS effect.However, approximately half FSR spectral shift between the type I and the type II are clearly shown.Without the CIFS effect, the spectral shift should be exactly half FSR.
It is worth noticing that the phase jump in φ 12 and φ 21 , if there is any, does not cause spectral shift, since the field undergoes both couple-in and couple-out that accumulates phase jump twice to yield 2π phase change.
For multi-ring add-drop device, where there are more than two gaps, the gap widths have to be arranged alternately between g L and g S in order to obtain the half FSR spectral shift.Multi-ring device of this type can be referred to as the generalized type II in contrast to the generalized type I in which all gap widths are g L or g S .
Notice that if there are more than one g C , referring to Fig. 2(c), g L and g S are not restricted to the opposite side of a particular g C , it is only required that one lies in the phase jump region and the other one lies in the region where there is no phase jump.

Application on wavelength interleaver
Based on the findings of the previous section, we propose an all-microring wavelength interleaver that consist a type I and a type II microring in series as shown in Fig. 9.An equal-spacing multi-channel input signal can be separated through the interleaver that has FSR twice the input We demonstrate the feasibility of the idea numerically through the following example: The spectra of the drop ports and the through port of the configuration in Fig. 9 can be calculated by the method as introduced in Appendix A.2; The elements of the S-matrix, including its magnitude and phase, for each half-ring coupler in Fig. 9 can be numerically measured as introduced in Sec.2.2, and then substituting into Eq.(A.14) and Eq.(A.15), applying the method twice, all the output spectra can be obtained.The values for the relevant parameters of the waveguide in Fig. 9 were the same as those for the waveguides in Sec.

Fig. 3 .
Fig. 3. Symmetrical half-ring coupler.(n core =2.5, n cladding =1.5, guide widths=300 nm, and α mat =0 dB/cm).nm,material loss coefficient α mat =0 dB/cm and outer radius R=12.15 μm.There was only one mode for the ring waveguide and it was a TE-like mode.The results are shown in Figs.4(a) and 4(b).There is a noticeable phase protruding about 0.1π in φ 11 right below the gap width of the phase jump, no satisfactory explanation could be given for the protrude phase.Nevertheless, it is clear that the phase jump indeed occurs in the half-ring coupler and its magnitude is close to π.The phase jump occurred at the minimum s 11 and maximum s 21 .There is only one phase jump for φ 11 and no phase jump for φ 21 in this case.The criteria for the phase jump to occur in the half-ring coupler could be modified from Eq.(10) as L e f f =n L c,e f f , where L e f f is the effective coupling length and L c,e f f is the effective Transfer Distance, and n=1 in this particular example.For half-ring coupler with larger radius of curvature with the same gap width, we expect that L e f f would be larger and L c,e f f would be smaller such that the number of phase jump would be increased as the gap width is varying.When the gap width and the radius of curvature for the half-ring coupler are fixed, the phase jump phenomenon will occur when the wavelength is varying.The simulation results are shown in Figs.5(a) and 5(b) for the same half-ring coupler but fixed gap width at 85 nm.The phase jump occurs at 1481.14 nm for this case.We define this wavelength as λ c .For longer wavelength, the evanescent field extends further out, a shorter L c,e f f is expected, and more phase jump would occur at longer wavelengths.The coefficient of K in Eq. (3) is inversely proportional to λ , the cosine and the sine function in Eq. (2a)-(2d) change sign whenever the

Fig. 7 .Fig. 8 .
Fig. 7. (a)I /I i with g L =350 nm.(b)I through /I i with g S =20 nm for single-arm microring.(Black line: calculated from the S-matrix method, Red line: calculated from the fulldomain FDTD method.) 2.3 with g L =350 nm and g S =20 nm.The results are shown in Figs.10(a)-10(c).It is apparent that the spectra of the drop ports were separated approximately half-way to each other.