Theory of polarization rotation and conversion in ... - IEEE Xplore

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 12, NO. 4, APRIL 2000

401

Theory of Polarization Rotation and Conversion in Vertically Coupled Microresonators Brent E. Little, Member, IEEE, and Sai T. Chu, Member, IEEE

Abstract—Two orthogonally polarized waveguide modes can exchange power through an intermediate level, if the polarization state of that level is tilted with respect to the polarization states of the waveguide. Microrings with modest sidewall angles support such tilted modes. Ultracompact, wavelength selective polarization rotators might be achieved. Simple analytic expressions are derived for polarization rotation by vertically-coupled resonators in terms of geometrical factors. Index Terms—Polarization rotation, resonators, wavelength filter, microcavity.

M

ANY devices in integrated optics are polarization sensitive, their response changes as a function of the input polarization state. Controlling the polarization of a lightwave signal in an integrated optical device is therefore of considerable importance. The polarization may need to be rotated, converted from one state to another, or the polarization states may need to be spatially separated. Integrated optic polarization converters include waveguides with periodic asymmetric loading [1], waveguides with periodic birefringence [2], adiabatically varying the birefringence axis of a crystal waveguide [3], polarization rotation in bending waveguides [4], [5], and polarization conversion based on coupling of higher order hybrid modes [6]. In this letter, we explore the concept of using tilted modes in resonators as a means to couple two orthogonal modes in an adjacent waveguide. Such devices many lead to very compact polarization rotators. Ultracompact ring resonators have been studied by numerous groups for their application in VLSI photonic circuits [7]–[13]. A ring resonator side-coupled to a single bus waveguide is depicted schematically in Fig. 1(a). It is assumed that the bus waveguide supports two orthogonal polarizations, which we label as the amplitude vectors Pe , and Ph for the TE e , and TM h modes, respectively. (Note however, that in general, the modes do not need to be pure TE or TM). The ring waveguide also supports one or more modes. We label the ring polarization state as Pr . The ring is resonant at particular wavelengths that satisfy the resonance condition. These resonant wavelengths depend on the ring modal effective index and ring radius. The ring resonator device will partially, or fully, convert an input polarization state Pe (or Ph ) prior to the ring, into state

()

()

Manuscript received November 23, 1999. B. E. Little is with the Laboratory for Physical Sciences, University of Maryland, College Park, MD 20740 USA and also with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. S. T. Chu is with the National Institute of Standards and Technology,Gaithersburg, MD 20899 USA. Publisher Item Identifier S 1041-1135(00)02878-0.

Ph (or Pe ) after the ring. For polarization conversion to occur, the polarization state of the ring mode Pr has to be tilted with respect to both Pe and Ph . The physical principle of polarization conversion or rotation can be understood qualitatively by referring to the sequence in Fig. 2(a)–(d). Consider an input signal initially in the polarization state Pe , which is depicted as a vector with a certain orientation in Fig. 2(a). This input signal will couple to the mode in the resonator, which has polarization state Pr , if Pe has a component in the same direction as Pr . This common component is depicted as the dashed line in Fig. 2(b). Once excited, the ring resonator mode can couple back out of the ring into the mode with polarization state Ph , if the mode Pr has a component in the direction of Ph . This common component is depicted as the dashed line in Fig. 2(c). The net output consists of the newly generated state Ph , shown in Fig. 2(d), and the power remaining in the input state Pe that was not converted by the ring. Due to orthogonality, modes Pe and Ph cannot directly exchange energy. The ring mode serves as an intermediate level through which power can be transferred. A useful correspondence can be made between the polarization converter in Fig. 1(a), and the ring resonator channel dropping filter depicted schematically in Fig. 1(b). The channel dropping filter is comprised of a ring resonator evanescently side-coupled to two bus waveguides. The ring may be vertically or laterally coupled to the bus guides. One bus waveguide serves as the input, while the other serves as the drop. Correspondence is achieved by associating the input signal with the input polarization state Pe; the drop signal with the orthogonal polarization Ph ; the coupling between the input bus and ring as the coupling between the input polarization Pe and the ring mode Pr e ; and the coupling between the drop bus and ring as the coupling between the orthogonal polarization Ph and the ring mode Pr h . Using this analogy, the analytic response of the polarization converter can be found from that of the channel dropping filter [7],

( )

( )

42e 2h e 41!2 + (2e + 2h )2 # "   4 2e 2h  o   arctan 1 0  = arctan (2e 0 2h)2 

 jjPPhjj =

(1a)

(1b)

where  is defined as the polarization conversion efficiency, o is the effective angular rotation of polarization at resonance, and ! is the frequency shift away from resonance. jPej and jPhj are the values of power for the Pe and Ph modes respectively. 2e and 2h are the fractions of power coupled from the ring into

1

1041–1135/00$10.00 © 2000 IEEE

402

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 12, NO. 4, APRIL 2000

P

P

P

Fig. 1. (a) Microring resonator polarization rotator. e is the input polarization state. h is the orthogonal state. r is the polarization vector of the ring mode. e and h are the amplitude coupling factors between the modes e and h and the mode r , respectively. (b) Equivalent channel dropping filter.

P

P

P

Fig. 2. Qualitative principle of polarization rotation in a ring resonator. Orthogonal waveguide polarizations e and h can exchange power by means of the intermediate state r in the ring.

P

P

P

the bus waveguide modes Pe and Ph respectively. It is assumed in the foregoing equations that loss in the ring is negligible. Loss can be incorporated in a straightforward manner [7]. The amplitude coupling coefficients e and h can be found from knowledge of the mode field patterns [7], [14],

e;h

= !"4

o

1E

Z

exp[0j1 z]

Z

e;h

z

e;h

@x @y @z

( ;)

(" 0 " )E r

x;y

g

r

(2)

( ;)

where "r x; y z and "g x; y z are the relative permitivitty distributions of the ring waveguide and bus waveguide in isolation, (that is, with the alternate waveguide replaced by uniform cladding). "o is the permittivity of free space, and ! is the radian frequency. Er x; y z and Ee;h x; y z are the power-normalized vector electric field distributions of the ring waveguide mode and the bus waveguide modes in isolation, respectively. e;h e;h 0 r is the propagation constant mismatch between the ring mode and the bus modes. Integration is carried out over the interaction region. The optical bandwidth over which rotation occurs is given approximately by the resonance e h Vg =R, where Vg is the group vebandwidth !3 dB locity of the ring mode, and R is the ring radius. The coordinate system x; y; z is depicted in Fig. 1.

( ;)

1

=

1

(

=

)

( ;)

Fig. 3. Cross section of a vertically coupled ring resonator. The ring has sloped sidewalls which leads to modes in the ring with tilted, or hybrid polarization. TM and TE components of the hybrid mode when  = 10 , and R = 20 m, are also shown.

The degree of rotation is controlled by the ratio of e to h as can be seen from (1b). The ratio of e to h can be changed by varying the tilt of the polarization Pr with respect to Pe and Ph . Polarization tilt can be controlled in several ways. These include choosing the waveguide width-to-height aspect ratio appropriately, using birefringent materials, or by applying an etch process to yield sloped sidewalls (either flared or undercut). Because the mode in a ring waveguide is pushed towards the outer wall, a sloped sidewall is an effective means of obtaining tilted hybrid modes. Consider a bus guide vertically coupled to an air-clad ring resonator [8], [12]. The cross-sectional geometry is depicted in Fig. 3, where R is the ring radius. The geometry is as follows: wr : m, hr : m, hg m, wg : m, ng : ; nr : ; ncl : ;t : m, and the resonant wavelength is approximately 1.55 m. The sidewall tilt is

= 15 = 1 65

= 15 =1 = 1 65 = 1 45 = 0 5

= 15

LITTLE AND CHU: THEORY OF POLARIZATION ROTATION AND CONVERSION IN VERTICALLY COUPLED MICRORESONATORS

Fig. 4. Polarization rotation angle as a function of sidewall slope  , for rings of radius 20 m and 40 m.

403

The foregoing analysis has shown that sloped sidewalls in a microring lead to coupling of orthogonal polarizations in the adjacent waveguide. Such polarization coupling can be detrimental to the operation of the conventional channel dropping filter depicted in Fig. 1(b). The efficiency of the conventional filter would be reduced because an input signal would couple energy into three other waves (the two polarizations in the output guide, and the orthogonal polarization in the input guide), rather than to a single output wave. In conclusion, hybrid or tilted modes in a microring resonator serve as an intermediate level through which orthogonal modes in a side-coupled waveguide can exchange power. Tilted resonator modes are achieved by sloped waveguide sidewalls, or by choosing the width-to-height aspect ratio appropriately. It is predicted that large rotation angles can be achieved by very compact structures. Sidewall slopes of 10 can lead to 45 polarization rotation. Because the polarization rotation is frequency selective, single waveguide wavelength filters might be realized by use of a polarization beam splitter placed after the rings. On the other hand, the analysis also shows that straight sidewalls are essential for high channel drop efficiencies in conventional dual-waveguide microring resonator filters. REFERENCES

Fig. 5. Polarization rotation angle as a function of ring waveguide width. The ring has a radius of 40 m, and the geometry is the same as that used for Fig. 4, (with  = 0).

given by  , as shown. The TE and TM field components of the hybrid mode circulating in the ring when the sidewall slope is 10 and the ring radius is 20 m, are also shown. These fields were calculated using a full vector numerical mode solver [15]. Although the mode is predominantly TM, approximately 20% of the energy is in the TE component in this case. Fig. 4 shows the predicted polarization rotation angle as a function of sidewall tilt angle  , for ring radii of 20 and 40 m. Negative values of  imply a waveguide which is undercut by the etch process. Large rotation angles are seen to be possible for moderate sidewall slope. The net rotation angle can be increased by cascading several rings along the bus waveguide. Hybrid resonator modes can also be realized by choosing the width-to-height aspect ratio appropriately. Fig. 5 plots the polarization rotation as a function of ring waveguide width wr . All other parameters previously given in Fig. 4 remain the same. The vertically coupled bus waveguide remains centered below the ring guide for all values of wr . Complete polarization conversion is possible over a very limited aspect ratio. In this case, the range of waveguide widths for which strong polarization rotation occurs is 50 nm. This corresponds to the transition region in which the ring mode evolves from predominantly TM polarized to predominantly TE polarized.

[1] Y. Shani et al., “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett., vol. 59, pp. 1278–1280, 1991. [2] R. Alferness et al., “Waveguide electro-optic polarization transformer,” Appl. Phys. Lett., vol. 38, pp. 655–657, 1981. [3] M. C. Oh et al., “Wavelength insensitive passive polarization converter fabricated by poled polymer waveguides,” Appl. Phys. Lett., vol. 67, pp. 1821–1823, 1995. [4] C. van Dam et al., “Novel compact polarization converters based on ultra-short bends,” IEEE Photon. Technology. Lett., vol. 8, pp. 1346–1348, 1996. [5] W. W. Lui, T. Hirono, K. Yokoyama, and W.-P. Huang, “Polarization rotation in semiconductor bending waveguides: A coupled-mode theory formulation,” J. Lightwave Technol., vol. 16, pp. 929–936, 1998. [6] K. Mertens et al., “First realized polarization converter based on hybrid supermodes,” IEEE Photon. Technol. Lett., vol. 10, pp. 388–390, 1998. [7] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol., vol. 15, pp. 998–1005, 1997. [8] B. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett., vol. 10, pp. 323–325, Mar. 2000. [9] D. V. Tishinin, P. D. Dapkus, A. E. Bond, I. Kim, C. K. Lin, and J. O'Brien, “Vertical resonant couplers with precise coupling efficiency control fabricated by wafer bonding,” IEEE Photon. Technol. Lett., vol. 11, pp. 1003–1005, 1999. [10] B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, “Vertically coupled glass microring resonator channel dropping filters,” IEEE Photon. Technol. Lett., vol. 11, pp. 215–217, 1999. [11] D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, and S. T. Ho, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6-nm free spectral range,” Opt. Lett., vol. 22, pp. 1244–1246, 1997. [12] S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett., vol. 11, pp. 691–693, 1999. [13] J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, Higher order filter response in coupled microring resonators, in IEEE Photon. Technol. Lett., to be published. [14] H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol., vol. LT-5, pp. 16–23, 1987. [15] Optical Waveguide Mode Solver Suite. Kitchner, ON, Canada: Apollo Photonics Inc..