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Dec 1, 2004 - Polarization conversion in ring resonator phase shifters. Andrea Melloni. Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via ...
December 1, 2004 / Vol. 29, No. 23 / OPTICS LETTERS

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Polarization conversion in ring resonator phase shifters Andrea Melloni Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy

Francesco Morichetti CoreCom, Via G. Colombo 81, Milano 20133, Italy

Mario Martinelli Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy, and CoreCom, Via G. Colombo 81, Milano 20133, Italy Received June 16, 2004 The effect of the polarization rotation induced by curved waveguides on the spectral behavior of phase shifter ring resonators is investigated both theoretically and experimentally. At resonance the polarization rotation that takes place in curved waveguides is strongly enhanced. The effect can be detrimental, or it can be exploited for new devices. The ring vectorial transfer function is derived, together with the conditions for the total conversion of TE polarization into TM polarization. These conditions are verified experimentally. © 2004 Optical Society of America OCIS codes: 130.3120, 230.5440, 230.5750, 260.5430, 230.7390, 130.1750.

Polarization conversion in dielectric waveguides takes place due to the interaction of dominant and nondominant components of the quasi-TE and quasi-TM modes. The curvature, the high refractive-index contrast, and the waveguide sidewall angle increase the hybridness of the modes and hence the conversion. Moreover, in ring-resonator-based devices the polarization rotation is enhanced near the resonances proportionally to the finesse of the cavity, affecting the spectral behavior of the device.1,2 The aim of this Letter is to model the single-ring phase shifter shown in Fig. 1(a) taking into account the effect of the polarization coupling occurring in the ring waveguide. Further aims are to compare the theoretical and experimental transfer functions, point out the detrimental effects, and investigate how to exploit the input –output wavelength-dependent polarization rotation. The polarization rotation in bent waveguides is largely discussed on a numerical basis in Refs. 3– 5, in which it is shown that a tightly bent waveguide with an index contrast of a few percent can induce considerable polarization coupling between the two fundamental quasi-TE and quasi-TM modes of the straight waveguide (thereinafter referred to as TE and TM modes). A simple way to describe the propagation along the ring waveguide is to resort to coupled-mode theory, according to which the TE and TM modes can be expressed as a combination of the two uncoupled hybrid modes of the bend. The transmission matrix relating the coupled-mode complex amplitudes from the beginning to the end of the ring waveguide is ∑ ∏ cos f 2 jR sin f 2jS sin f , (1) Tb 苷 z21 2jS sin f cos f 1 jR sin f where z21 苷 g exp共2j bLr 兲, Lr is the length of the ring, g is the round-trip loss, b 苷 共bTE 1 bTM 兲兾2 is the aver0146-9592/04/232785-03$15.00/0

age phase constant of the two modes, R 苷 Db兾2d, S 2 苷 1 2 R 2 , and f 苷 dLr is a measure of the induced polarization rotation. The two modes exchange power with spatial period LB 苷 p兾2d, and the maximum power coupling ratio is KpM 苷 1 2 Db 2 兾4d 2 , the birefringence being Db 苷 bTE 2 bTM , where d 苷 共Db 2 兾4 1 kp 2 兲1兾2 and kp is the field coupling coeff icient between the two modes.1,5 By use of this model, the vectorial transfer function of the ring phase shifter is obtained. Under the hypothesis of a polarization-independent coupler, the TE–TE (Hee ) and TM– TM (Hmm ) ring transfer functions are given by ee 苷 H mm

rz22 2 共1 1 r 2 兲z21 cos f 6 jRt2 z21 sin f 1 r , 1 1 r 2 z22 2 2rz21 cos f

(2)

where 2jt is the f ield coupling coefficient of the directional coupler and r is its transmission coefficient [see Fig. 1(a)]. Similarly, the cross-polarization transfer functions TE– TM and TM– TE are 苷 H em me

jSt2 z21 sin f . 1 1 r 2 z22 2 2rz21 cos f

(3)

Fig. 1. (a) Single-ring phase shifter. (b) Loci of poles and zeros of Eq. (2) for a phase shifter with small losses and Db 苷 0. © 2004 Optical Society of America

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In the absence of both polarization coupling and birefringence, S, f, and Hem vanish and Hee assumes the well-known expression6 Hee 苷 共r 2 z21 兲兾共1 2 rz21 兲 with one pole and one zero. In the case of polarization coupling, instead, the two poles common to both Hee and Hem split in complex-conjugate positions zp 苷 gr exp共6jf兲 and two zeros appear in Hee . The filter shows a second-order spectral response and, using a microwave term, it is a dual-mode filter because two coupled modes resonate in the same physical cavity. Dual-mode filters are commonly used in microwaves because of their compactness. As a f irst investigation let us consider the simple case of a ring phase shifter without birefringence. The TE and TM modes resonate at the same frequencies, S 苷 1, R 苷 0, and maximum power coupling KpM is unitary. Equations (2) and (3) simplify, and the loci of the poles and zeros are shown in Fig. 1(b) as a function of f. When the polarization coupling is increased from f 苷 0, the two zeros approach, overlap, and become complexly conjugated. If r , g (maximum phase shifter), one zero lies outside the unitary circle, and hence there always exists a value fcr for which Hee (Hmm ) vanishes and the input TE (TM) signal is totally converted to TM (TE). More generally, the lossless phase shifter acts as a wave plate angled by 45± with a rotating power equal to pf兾fcr . The critical coupling coefficient that brings the zero onto the unit circle and hence guarantees the total conversion is cos fcr 苷

r共1 1 g 2 兲 . g共1 1 r 2 兲

20 dB, an interesting characteristic for polarization interleaved wavelength-division multiplexing systems. When Db fi 0, the dual-mode resonator acts similarly, but the interplay of the ring parameters is more complex. For brevity we consider only a phase shifter with t2 苷 0.3, g 苷 0.25 dB兾turn, a FSR of 100 GHz, and a birefringence equal to 1024 . Figure 3 shows both the Hee and Hem intensity transfer functions for polarization coupling Kp equal to 0%, 5%, and 10% per turn. In the case of polarization rotation, two notches appear in Hee , corresponding to the Hem resonant peaks, and a polarization conversion takes place. Increasing the coupling causes the two peaks to tend to separate as in the Db 苷 0 case, but they never reach the total conversion because of the phase mismatch between the two coupled modes. In the figure the position of the TE and TM resonances for Kp 苷 0 are indicated by marks on the abscissa. These phenomena have been experimentally observed on ring resonators realized in silicon oxynitride technology. The waveguide, described in Ref. 7, is rib shaped, has a 6% of index contrast, and supports a bending radius of 200 mm with negligible radiation losses. As an example, in this waveguide a 274-mm bending radius curve induces a maximum TE–TM

(4)

The spectral response over a free spectral range (FSR) of a phase shifter with Db 苷 0, t2 苷 0.5, and g 苷 1 is shown in Fig. 2 for values of f equal to 0.05, 0.34共fcr 兲, and p兾2, corresponding to a polarization coupling Kp per turn equal to 0.25%, 11.1%, and 100%, respectively. The resonant frequencies depend on polarization coupling f, and the output state of polarization is wavelength dependent. For f , fcr , only a partial conversion takes place, whereas for f $ fcr , total conversion can occur and the resonances split into two separate peaks. These correspond to the resonances of the two hybrid modes of the ring, whose phase constants bb e are modif ied by m the coupling as bb me 苷 b 6 d. When f is increased, the poles continue to separate, and for f 苷 p兾2 the transfer function is symmetrical with half the FSR of the original structure. The TE– TE (TM – TM) response presents a notch at each resonant frequency, whose width depends on t and f. For f 苷 fcr a transmission zero appears and the two poles give rise to a second-order maximum f latness response. The conversion, def ined at 21 dB, occurs over a frequency range of B 苷 FSRt2 兾共2pr兲, with an extinction ratio of ER 苷 t8 兾共1 1 6r 2 1 r 4 兲2 . Both relations are derived from Eq. (3). In this case the single-ring shifter operates as a periodic polarization converter. As an example, a 100-GHz ring shifter with a large coupling coefficient of t2 苷 0.6 guarantees a bandwidth of 15 GHz and an extinction ratio of

Fig. 2. jHee j2 (solid curve) and jHem j2 (dashed curve) of lossless phase shifters with t2 苷 0.5 and Db 苷 0 for three values of f. fcr 苷 0.34.

Fig. 3. jHee j2 and jHem j2 of phase shifters with t2 苷 0.3, g 苷 0.25 dB兾turn, a FSR of 100 GHz, birefringence of 1024 for three values of polarization coupling.

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Fig. 4. Comparison between measured (solid curve) and simulated (dashed curve) jHee j2 and jHem j2 of phase shifters with a FSR of (a) 100 GHz and (b) 50 GHz. The position of the resonances in the absence of polarization coupling is shown.

coupling KpM 苷 1% after a distance LB 兾2 苷 1.4 mm but increases to more than 5% in the case of a sidewall angle equal to 10±. Figure 4(a) shows the spectral response over a single FSR of a phase shifter with FSR 苷 100 GHz. The bending radius is 274 mm, and the input mode is polarized quasi-TE. Both measured and theoretical Hee and Hem intensity responses are shown, and the agreement is fairly good. By f itting the measured data, the following parameters have been retrieved: losses of 0.75 dB兾turn, polarization-independent coupling coefficient t2 苷 0.38, birefringence Db 苷 1.1 3 1024 , and kp 苷 0.85Db 苷 380 m21 . Far from the resonances the weak conversion is lower than 225 dB. At the resonances, instead, the polarization conversion is strongly enhanced and a considerable amount of TM is produced. The two jHem j2 peaks reach the same level because the directional coupler is polarization independent. Instead,

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jHee j2 presents two notches with different depths. The deeper notch occurs at the resonance of the hybrid mode closer to the resonant frequency of the input mode. In our case the quasi-TE mode resonates on the right and the quasi-TM mode on the left of the central frequency. In the case of a TM-polarized input field, the deeper notch would be the left one, as predicted by Eq. (2). Finally, it is intriguing to note that, thanks to the ring losses, at the resonant frequency of the quasi-TE mode an input TE field is almost completely converted into the TM polarization, whereas an input TM f ield remains almost TM. Figure 4(b) shows the experimental spectral response in the case of a TE input f ield of a phase shifter with a FSR of 50 GHz. In this case a much stronger conversion, almost total at resonance, is observed. The f itting yields the following parameters: losses 0.21 dB兾turn, t2 苷 0.22, Db 苷 0.5 3 1024 , and kp 苷 1.98Db 苷 400 m21 . The higher value of kp , responsible for the separation of the two resonances by half a FSR (d 艐 kp ), is probably due to a higher slope of the waveguide sidewalls. In conclusion, we have experimentally observed the polarization conversion in ring resonators and a simple model that fairly well agrees with measured spectral responses is proposed. These phenomena can be exploited in polarization converter devices, polarization interleaved transmission systems, or in other polarization-sensitive components. On the other hand, these phenomena can be detrimental in interferometric devices that include phase shifters to modify the phase response, such as in ring-loaded Mach– Zehnder filters or dispersion-compensating devices, and a careful investigation is in progress. A. Melloni’s e-mail address is [email protected]. References 1. B. E. Little and S. T. Chu, IEEE Photon. Technol. Lett. 12, 401 (2000). 2. M. C. Larciprete, E. J. Klein, A. Belardini, D. H. Geuzebroek, A. Driessen, and F. Michelotti, in Microresonators as Building Block for VLSI Photonics, M. Bertolotti, A. Driessen, and F. Michelotti, eds. (American Institute of Physics, New York, 2004). 3. S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, IEEE Photon. Technol. Lett. 13, 681 (2001). 4. N. Somasiri and B. M. A. Rahman, J. Lightwave Technol. 21, 54 (2003). 5. W. W. Lui, T. Hirono, K. Yokoyama, and W. Huang, J. Lightwave Technol. 16, 929 (1998). 6. C. K. Madsen and J. H. Zhao, in Optical Filter Design and Analysis (Wiley, New York, 1999). 7. A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli, Opt. Lett. 28, 1567 (2003).