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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 12 — Jun. 13, 2005
  • pp: 4580–4588
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Phase engineering for ring enhanced Mach-Zehnder interferometers

S. Darmawan, Y. M. Landobasa, and M. K. Chin  »View Author Affiliations


Optics Express, Vol. 13, Issue 12, pp. 4580-4588 (2005)
http://dx.doi.org/10.1364/OPEX.13.004580


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Abstract

Ring resonators are waveguide realizations of Fabry-Perot resonators which can be readily integrated in array geometries to implement many useful functions. Its nonlinear phase response can be readily incorporated into a Mach-Zehnder interferometer to produce specific intensity output function. We present two generalized array configurations of ring-coupled MZI and discuss their characteristics in terms of the amplitude and phase response of the ring arrays as well as the transmission output of the MZIs. The two types of array have distinct transfer functions and effective phase shifts, and can be tailored to phase-engineer a wide-range of MZI transmission functions.

© 2005 Optical Society of America

1. Introduction

Mach-Zehnder interferometer (MZI) provides an efficient means for converting phase modulation to intensity modulation. In this device, a single input wave is split between two arms, and a phase shift is induced either in a single arm or in both arms in a push-pull manner. There could be a single output or two complementary outputs depending on whether the two arms are combined via a Y branch or a 2×2 3dB coupler. The phase difference could be induced by an electro-optic or all-optical effect, a change in temperature or one of many other possible control parameters. As such, the MZI is a versatile device with many applications, for examples, as a space switch [1

1. John E. Heebner and Robert W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt Lett. 24, 847–849, June 1999. [CrossRef]

2

2. Li Chun-Fei and Bananej Alireza, “Finesse-enhanced ring resonator coupled Mach-Zehnder Interferometer all-optical switches,” Chin. Phys. Lett. 21, 90–93 (2004). [CrossRef]

], intensity modulator [3

3. L. Liao, D. Samara-Rubio, M. Morse, Ansheng Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express. 13, 3130–3135 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3129. [CrossRef]

], optical filter or sensor [4

4. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001). [CrossRef]

,5

5. C.-Y. Chao and L. J. Guo,“A new interferometric sensor with ring-feedback MZI”, in Proceedings of IEEE on Sensors. 1, 569–572, Oct. 2003.

].

To engineer the performance of MZI, one could tailor the phase shift by introducing different passive or active building blocks into the MZI. One effective way is by coupling a ring resonator (RR) to one of the MZI arms. Since the ring resonator is a resonant structure, the phase accumulated inside the ring will enhance the phase difference between the MZI arms. We shall generally call this MZI a ring-enhanced MZI (REMZI). In this paper, we study the effect of introducing more complicated ring resonator configurations, in order to understand how far we can tailor the phase shift and the resulting MZI transfer function. The higher-order ring resonator structures are also filters [7

7. J. E. Heebner, V. Wong, S. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726–730 (2004). [CrossRef]

] and have unique transmission and phase responses depending on the configurations. When coupled to MZI they greatly enhance the sensitivity of the MZI to frequency and effective index changes. In the most general case, we will consider a 2-D array of RRs, coupled to the MZI in two possible configurations to be discussed below. The resonators will at first be assumed lossless so as to focus our study solely on the phase shift introduced by the ring structures. We will then discuss the effect of loss.

2. Ring-enhanced MZI

In a REMZI, the ring resonator structure is coupled to one arm of an MZI. The outputs of the MZI can be expressed in general by the following matrix relation:

[EbarEcross]=[riκiκr][Texp(iΔφeff)00exp(iΔφb)][riκiκr][Ein0]
(1)

where the first and the third matrices on the right-hand side represent the output and input couplers of the MZI [7

7. J. E. Heebner, V. Wong, S. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726–730 (2004). [CrossRef]

,9

9. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express. 12, 90–103 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-90. [CrossRef] [PubMed]

], respectively, and T exp(Δφeff)≡teff in the second matrix summarizes the complex transfer function (amplitude and phase) of the ring structure which is coupled to the upper arm of the MZI. For generality we have included an additional phase bias Δφb (active or passive) at the lower MZI arm. However, we will assume that Δφb=0 so as to focus on the phase shift introduced by the ring structures. To simplify the analysis, all rings and ring couplers are assumed to be identical, and for the input and output couplers 3-dB coupling ratios are assumed (i.e., r=κ=1/√2). From (1), the bar and cross output powers are given by

Pb=Ebar2=(1+T22TcosΔφeff)4
Pc=Ecross2=(1+T2+2TcosΔφeff)4
(2)

When T=1, (2) reduces to the usual equation Pb=sin 2φ/2) and Pb=cos 2φ/2) for a symmetric MZI. The two outputs are complementary.

Fig. 1. The effective transmission phase shift versus the ring single-pass phase shift for a single ring coupled to one waveguide (red), and to two waveguides (blue).

3. Side-Coupled Ring Enhanced Mach-Zehnder Interferometer (SC-REMZI)

Figure 2 shows the generalized 2D array SC-REMZI configuration, which consists of columns of rings side-coupled to the upper MZI arm. The rings are mutually coupled in the same column, but not between columns.

Fig. 2. Side Coupled REMZI (inter-resonator coupling occurs within the columns, but not between the columns).

The mutually coupled ring structure is known as a coupled-resonator optical waveguide (CROW) [6

6. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

], through which light can propagate when the rings are on resonance. Being an extension of the single-ring all-pass filter, the effective phase shift in each ring is 2π, and the total phase shift for M rings is M×2π. However, because of mutual coupling, the resonance frequency shows M-fold splitting, and the accumulated phase shift increases with M in a staircase manner. When coupled to the MZI, this ripple-like phase shift is converted into an intensity pattern with M sharp oscillations in the transfer function, as shown in Fig. 3(a). Note that the split peaks (in red) are sharper relative to the original peak of the one-ring case (blue curve), and the spacing between them is determined only by the coupling coefficient between the resonators.

In the case of a single row of N identical rings which are coupled to one arm of the MZI, but not mutually coupled to each other, we merely have a series of all-pass filters each accumulating a phase shift of 2π across a resonance, giving a total phase shift of N×2π. Hence, the MZI transfer function shows N oscillations, with the most rapid change occurring at the resonance, as shown in Fig. 3(b).

Fig. 3. SC-REMZI: Ring effective phase (upper curve) and the bar output transmission (lower curve) as a function of the ring round trip phase δ for (a) a single column of M (=1,2,3) resonators; (b) a single row of N (=1,2,3) resonators, and (c) N×M 2D arrays (lossless case a=1 ; r=0.8). The insets show the ring and REMZI configurations. Since T=1 throughout, the amplitude responses are not shown.

4. Coupled Ring Enhanced Mach-Zehnder Interferometer (C-REMZI)

Fig. 4. Coupled REMZI (inter-resonator coupling occurs within the columns, but not between the columns).

Let us first consider the case of a single column of M resonators, where M is an odd integer. When the resonators are on resonance, input light will propagate through the array and be directed back into the MZI. Otherwise, the incident light will be reflected by the array and is assumed lost at the end of the input bus waveguide. This case has been demonstrated in [12

12. George T. Paloczi, Yanyi Huang, and A. Yariv,“Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express. 11, 2666–2671 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2666. [CrossRef] [PubMed]

]. Figure 5(a) shows the sharp transmission bands when the resonators are on resonance. The M ripples in the passband are due to the mutual coupling between the M resonators. Similarly, the phase response shows a ripple pattern and the total phase shift is , since the effective phase shift is π per ring in this case. The MZI output shows corresponding oscillatory behavior within the resonance bands. Outside the resonance band, the phase shift is zero since T=0. Here the output at the bar port is a constant 0.25 which is the fraction of light that has passed through the two 3-dB couplers. The total power is conserved when both the bar and cross output powers and the reflected power are accounted for.

Fig. 5. C-REMZI: Transmission amplitude T (top curve), effective phase e Δφeff (middle) and the bar output transmission (bottom), in the absence of loss (a=1), as a function of the ring round trip phase δ, for (a) a single column of M (=1,3,5) resonators; (b) a single row of N (=1,3,5) resonators; (c) N×M 2D arrays (r=0.8). The insets show the ring and the REMZI configurations.

In the case of the 1-D side-coupled array with N resonators (refer to Fig. 5(b)), light cannot propagate through the array when the resonators are on resonance, but is instead reflected in the opposite direction and re-directed into the MZI via the other bus waveguide. Furthermore, because the double channels provide many paths for the waves to feedback to earlier resonators, the array is similar to a distributed feedback grating, giving rise to Bragg resonances when the Bragg condition, λj=2neffLb/j(j=1,2,3…), is satisfied [10

10. John E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics”, J. Opt. Soc. Am. B 21, 1818–1832 (2004). [CrossRef]

]. The transfer function, therefore, shows two types of resonances in general, one that depends on the resonator cavity length (L c) and another that depends on the resonator spacing (L b). These two resonances, however, overlap if we set 2L b=L c. This is the case shown in Fig. 5(b) and (c). In this case, the reflection bands may be considered as the photonic bandgaps (PBG) of the periodic structure. For an array of N rings, there are N zero crossings (or N-2 sidelobes) between two adjacent bands. These sidelobes are due to interference between the reflected waves from the resonators.

The 2-D array can be analyzed column-wise first, then row-wise. Column-wise, each column can be reduced to an equivalent resonator represented by an effective transfer matrix. The 2-D array is then reduced to a 1-D row array of equivalent resonators, hence the phase response is similar to that discussed above. However, because embedded in each equivalent resonator is a CROW with its PBG structure, the sidelobes in the array transmission spectrum is significantly suppressed compared to the 1×N case [11

11. Y. M Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” submitted to IEEE J. Quantum Electron.

]. In response, the ripples outside the resonance band in the MZI output are much more subdued.

5. The effects of coupling and loss

It is convenient to study the impact of the ring resonator on the MZI by varying the coupling coefficient κ=1r2, where r is the reflectivity coefficient at the ring coupling region. For example, when r=1, the ring resonator structure is completely decoupled from the MZI. On the other hand, if r=0, then the rings become a physical extension of the MZI arm, and the phase shift generated in the rings is proportional to the path length. Consider for example a 3x3 array: the light will propagate through all 9 rings sequentially in the SC-REMZI case, but will propagate only through 3 rings in the C-REMZI case, hence the effective phase shifts are linear and total 18π and 3π, respectively, across a resonance as shown in Fig. 6(a).

The corresponding situation when r=0.8 is shown in Fig. 6(b). It is clear that increasing the reflectivity r has a band-limiting effect in that the MZI response is compressed to a band with a bandwidth determined by r. The higher the reflectivity (the lower the coupling), the narrower will be the transmission band and the more nonlinear the phase shift, and hence the more resonant will be the device output. The number of resonances in the output depends on how many rings are effectively coupled to the MZI arm. In this regard the side-coupled configuration is more effective, whereas in the C-REMZI case some of the rings in the array are quite redundant.

Fig. 6. The effect of coupling coefficient, or reflectivity (r), on the ring phase and the MZI bar output for the two ring array configurations. (a) r=0, (b) r=0.8.

However, one must also consider another important effect that increases with the number of effectively coupled rings, that is, the response time of the device, which is the time it takes for an optical pulse to propagate through the large number of rings. The latency in a ring is proportional to the resonator’s finesse which increases with increasing r. Hence, the more resonant the device output, the higher will be the latency through all the rings. Clearly if the latency exceeds the optical pulse width then there will be no interference effect at the MZI output. Hence, the long response time of a large array of micro-rings will limit the device bandwidth and the signal bit rate that can be passed through it.

The array size is not only limited by the response time consideration but also by the propagation loss in the micro-rings. The ring loss is defined using the round-trip attenuation factor a (for a lossless ring, a=1). The effects of loss on the two REMZI configurations are shown in Fig. 7. It can be seen that even a 5% round trip loss has a detrimental effect on device performance, especially in the SC-REMZI configuration where the light propagates through all the rings. Since the signal must pass through all the rings, all the rings must have precisely overlapping resonant frequencies, identical coupling coefficients, and very low loss. Therefore, fabrication requirement is stringent, and these REMZI designs are practical only if very low loss and uniform rings can be realized. Recently, very high order multi-ring filters with very high Q resonators have been realized using low-loss Hydex material [14

14. 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). [CrossRef]

], showing the technological possibility of realizing large and uniform resonator arrays.

Fig. 7. The effect of ring loss on the 3×2 Array (a) SC-REMZI and (b) C-REMZI configurations.

6. Discussion

We have presented in this paper two generalized array configurations of ring-coupled MZI, namely SC-REMZI and C-REMZI, and discussed their characteristics in terms of the amplitude and phase response of the ring arrays as well as the transmission output of the MZIs. Because the rings are resonant devices, they greatly enhance the sensitivity of the MZI to frequency and effective index changes. In the side-coupled (SC) structure, the array is coupled to waveguide only on one side and hence the wave can only propagate in the forward direction. In the coupled-REMZI case, the array is coupled on both sides to waveguides so there is optical feedback between the resonators. Because of this fundamental difference the two types of array have distinct transfer functions and effective phase shifts, which can be tailored to phase-engineer a wide-range of MZI transmission functions.

From the application point of view, the side-coupling configuration is probably better as it can be designed to give any number of sharp resonances within a small band in the MZI output, which may be a desirable feature for some nonlinear switching and sensing applications [4

4. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001). [CrossRef]

,5

5. C.-Y. Chao and L. J. Guo,“A new interferometric sensor with ring-feedback MZI”, in Proceedings of IEEE on Sensors. 1, 569–572, Oct. 2003.

]. The 2-D array coupled between two waveguides, on the other hand, has been shown to form a near-ideal bandpass filter characterized by a flat-top, square and ripple-free amplitude response and a largely linear phase response, if the array is sufficiently large (e.g., M=3 and N=10). This characteristic is attributed to the 2-D nature of the photonic bandgap exhibited by the 2-D periodic structure [11

11. Y. M Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” submitted to IEEE J. Quantum Electron.

]. Despite the large number of rings, the phase response is not greatly enhanced and is approximately linear except near the bandedges. For this reason this 2-D ring structure does not significantly modify the MZI output. Of course, MZI is not the only way to utilize the phase response of these ring resonator structures, but it is probably the simplest and the most effective device. In this study, we have shown the amplitude and phase characteristics of various ring configurations and how they can be used to physically engineer the MZI transmission behavior. We have also discussed some of the practical issues involved, including the effects of loss, delay time, and fabrication requirement.

References and links

1.

John E. Heebner and Robert W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt Lett. 24, 847–849, June 1999. [CrossRef]

2.

Li Chun-Fei and Bananej Alireza, “Finesse-enhanced ring resonator coupled Mach-Zehnder Interferometer all-optical switches,” Chin. Phys. Lett. 21, 90–93 (2004). [CrossRef]

3.

L. Liao, D. Samara-Rubio, M. Morse, Ansheng Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express. 13, 3130–3135 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3129. [CrossRef]

4.

R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001). [CrossRef]

5.

C.-Y. Chao and L. J. Guo,“A new interferometric sensor with ring-feedback MZI”, in Proceedings of IEEE on Sensors. 1, 569–572, Oct. 2003.

6.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

7.

J. E. Heebner, V. Wong, S. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726–730 (2004). [CrossRef]

8.

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photon. Technol. Lett. 12, 810–812 (2000). [CrossRef]

9.

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express. 12, 90–103 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-90. [CrossRef] [PubMed]

10.

John E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics”, J. Opt. Soc. Am. B 21, 1818–1832 (2004). [CrossRef]

11.

Y. M Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” submitted to IEEE J. Quantum Electron.

12.

George T. Paloczi, Yanyi Huang, and A. Yariv,“Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express. 11, 2666–2671 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2666. [CrossRef] [PubMed]

13.

G. Lenz, B.J. Eggleton, C. R. Giles, C. K. Madsen, and R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998). [CrossRef]

14.

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). [CrossRef]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(230.5750) Optical devices : Resonators

ToC Category:
Research Papers

History
Original Manuscript: May 10, 2005
Revised Manuscript: May 30, 2005
Manuscript Accepted: June 1, 2005
Published: June 13, 2005

Citation
S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4580


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References

  1. John E. Heebner, Robert W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt Lett. 24, 847–849, June 1999. [CrossRef]
  2. Li Chun-Fei, Bananej Alireza, “Finesse-enhanced ring resonator coupled Mach-Zehnder Interferometer all-optical switches,” Chin. Phys. Lett. 21, 90–93 (2004). [CrossRef]
  3. L. Liao, D. Samara-Rubio, M. Morse, Ansheng Liu, D. Hodge, D. Rubin, U. D. Keil, T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express. 13, 3130–3135 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3129. [CrossRef]
  4. R. W. Boyd, J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001). [CrossRef]
  5. C.-Y. Chao, L. J. Guo,“A new interferometric sensor with ring-feedback MZI”, in Proceedings of IEEE on Sensors. 1, 569–572, Oct. 2003.
  6. A. Yariv, Y. Xu, R. K. Lee, A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]
  7. J. E. Heebner, V. Wong, S. Schweinsberg, R. W. Boyd, D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726–730 (2004). [CrossRef]
  8. G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photon. Technol. Lett. 12, 810–812 (2000). [CrossRef]
  9. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express. 12, 90–103 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-90. [CrossRef] [PubMed]
  10. John E. Heebner, P. Chak, S. Pereira, J. E. Sipe, R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics”, J. Opt. Soc. Am. B 21, 1818–1832 (2004). [CrossRef]
  11. Y. M Landobasa, S. Darmawan, M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” submitted to IEEE J. Quantum Electron.
  12. George T. Paloczi, Yanyi Huang, A. Yariv,“Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express. 11, 2666–2671 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2666. [CrossRef] [PubMed]
  13. G. Lenz, B.J. Eggleton, C. R. Giles, C. K. Madsen, R.E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998). [CrossRef]
  14. B.E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004). [CrossRef]

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