## High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide

Optics Express, Vol. 16, Issue 6, pp. 4177-4191 (2008)

http://dx.doi.org/10.1364/OE.16.004177

Acrobat PDF (690 KB)

### Abstract

A novel electro-optic silicon-based modulator with a bandwidth of 78GHz, a drive voltage amplitude of 1V and a length of only 80*µ*m is proposed. Such record data allow 100Gbit/s transmission and can be achieved by exploiting a combination of several physical effects. First, we rely on the fast and strong nonlinearities of polymers infiltrated into silicon, rather than on the slower free-carrier effect in silicon. Second, we use a Mach-Zehnder interferometer with slotted slow-light waveguides for minimizing the modulator length, but nonetheless providing a long interaction time for modulation field and optical mode. Third, with this short modulator length we avoid bandwidth limitations by *RC* time constants. The slow-light waveguides are based on a photonic crystal. A polymer-filled narrow slot in the waveguide center forms the interaction region, where both the optical mode and the microwave modulation field are strongly confined to. The waveguides are designed to have a low optical group velocity and negligible dispersion over a 1THz bandwidth. With an adiabatic taper we significantly enhance the coupling to the slow light mode. The feasibility of broadband slow-light transmission and efficient taper coupling has been previously demonstrated by us with calculations and microwave model experiments, where fabrication-induced disorder of the photonic crystal was taken into account.

© 2008 Optical Society of America

## 1. Introduction

1. L. Liao, A. Liu, D. Rubin, J. Basak, Y. Chetrit, H. Nguyen, R. Cohen, N. Izhaky, and M. Paniccia, “40Gbit/s silicon optical modulator for high-speed applications,” Electron. Lett. **43**, 20072253 (2007). [CrossRef]

2. B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, “Electrooptic polymer ring resonator modulation up to 165GHz,” IEEE J. Sel. Top. Quantum Electron. **13**, 104–110 (2007). [CrossRef]

*r*

_{33}=10pm/V [3

3. D. Rezzonico, M. Jazbinsek, A. Guarino, O.-P. Kwon, and P. Günter, “Electro-optic Charon polymeric microring modulators,” Opt. Express **16**, 613–627 (2008), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-2-613. [CrossRef] [PubMed]

*r*

_{33}=170pm/V [4

4. Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, J. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, “Hybrid polymer/sol-gel waveguide modulators with exceptionally large electro-optic coefficients,” Nature Photonics **1**, 180–185 (2007). [CrossRef]

5. E. M. McKenna, A. S. Lin, A. R. Mickelson, R. Dinu, and D. Jin, “Comparison of *r*_{33} values for AJ404 films prepared with parallel plate and corona poling,” J. Opt. Soc. Am. B **24**, 2888–2892 (2007). [CrossRef]

6. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express **13**, 5216–5226 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14-5216. [CrossRef] [PubMed]

7. G. Wang, T. Baehr-Jones, M. Hochberg, and A. Scherer, “Design and fabrication of segmented, slotted waveguides for electro-optic modulation,” Appl. Phys. Lett. **91**, 143109 (2007). [CrossRef]

8. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature (London) **435**, 325–327 (2005). [CrossRef] [PubMed]

11. L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. **90**, 071105 (2007). [CrossRef]

12. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

13. J.-M. Brosi, J. Leuthold, and W. Freude, “Microwave-frequency experiments validate optical simulation tools and demonstrate novel dispersion-tailored photonic crystal waveguides,” J. Lightwave Technol. **25**, 2502–2510 (2007). [CrossRef]

*π*-voltage are discussed. Section 6 is devoted to performance data of an optimized integratable Mach- Zehnder modulator, and in Section 7 a taper is proposed for coupling efficiently to the slow-light mode of the phase modulator sections. The Appendix gives derivations of important relations.

## 2. The modulator

*E*oriented parallel to the substrate plane. Y-branches split and combine the signals. Both arms of the MZ interferometer comprise phase modulator (PM) sections of length

_{x}*L*. The PM sections consist of photonic crystal (PC) line defect (LD) waveguides (WG) with a narrow gap of width

*W*

_{gap}in the center, and are infiltrated with a

*χ*

^{(2)}-nonlinear material (Pockels effect). In such a PC-WG, the dominant optical field component

*E*is strongly confined to the gap (see Section 4). The nonlinear material is assumed to be poled [5

_{x}5. E. M. McKenna, A. S. Lin, A. R. Mickelson, R. Dinu, and D. Jin, “Comparison of *r*_{33} values for AJ404 films prepared with parallel plate and corona poling,” J. Opt. Soc. Am. B **24**, 2888–2892 (2007). [CrossRef]

*x*-direction; the associated electro-optic coefficient is

*r*

_{33}. Therefore, only the

*x*-components of the optical field and of the modulating field are relevant. The nonlinear material is assumed to respond instantaneously.

*n*

_{D}≈2×10

^{16}cm

^{-3}), to be sufficiently conductive (σ=10Ω

^{-1}cm

^{-1}) without introducing excessive optical loss. The edges of the PM silicon slabs are metalized with aluminum on top, and the three metallic layers (dark shading) running in parallel to the nonlinear optical WGs form a microwave coplanar waveguide (CPWG). The two outer electrodes are grounded, and a modulating voltage

*U*applied to the center electrode generates a voltage wave that travels along the line and drives the PM sections in push-pull mode. In each of the arms, the electric modulation field

*E*

_{el}=

*U*/

*W*

_{gap}is dominantly oriented along the

*x*-direction and almost completely confined to the gap. An optical quasi-TE field is launched at the input side of the MZM in Fig. 1. The phase shifts induced on the two MZI arms by the respective PMs are +ΔΦ=+[ΔΦ

_{0}+ΔΦ(

*t*)] and -ΔΦ=-[ΔΦ

_{0}+ΔΦ(

*t*)], where ΔΦ(

*t*)~

*U*(

*t*) represents the time-varying part. The bias voltage

*U*

_{0}is chosen such that the offset point is set to ΔΦ

_{0}=

*π*/4 (total phase difference 2ΔΦ

_{0}=

*π*/2 between both arms, MZ interferometer in quadrature). For switching the MZM from full transmission to full extinction, the phase in each PM section must be changed from ΔΦ=-

*π*/4 to ΔΦ=+

*π*/4, which corresponds to a voltage change from

*U*=-

*U*/4 to

_{π}*U*=+

*U*/4. The modulation voltage

_{π}*U*that is required to change the phase in one modulator arm by

*π*is called

*π*-voltage

*U*.

_{π}## 3. MZM optimization strategy

*f*

_{3dB}, require low modulation voltage amplitude

*U*, should be fabricated based on CMOS processes, and it should have small size for easy on-chip integration. The MZM modulation bandwidth is in general limited by

*RC*-effects and by the spatial walk-off between the electrical and optical waves. The spatial walk-off can be avoided by a travelling-wave design, where the group velocities

*v*

_{g, el}and

*v*

_{g, opt}of the electrical and optical waves are same, and where the CPWG is terminated with a matching impedance

*Z*

_{L}to avoid reflections.

## 4. Slow-wave phase modulator

*W*

_{gap}is cut out in form of a slot. The PC consists of a silicon slab with a triangular lattice of air holes having a lattice constant

*a*. For a

*W*line defect, a number of

_{i}*i*rows of holes are omitted. The width

*W*

_{1}of the resulting waveguide, see Fig. 2(b), depends on

*i*, which need not be an integer. As explained previously, the silicon structure is covered with a highly nonlinear poled electro-optic polymer, which fills both the slot and the PC holes. Due to the high index-contrast between silicon (

*n*

_{Si}=3.48) and polymer (

*n*

_{poly}=1.6), the optical quasi-TE mode is mainly confined to the polymer-filled gap, Fig. 2(b) and [14

14. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**, 1209–1211 (2004). [CrossRef] [PubMed]

*U*we need to avoid any voltage drop in the silicon material between the electrodes (Al in Fig. 2(a)) and the gap. To this end, the silicon must be made sufficiently conductive by doping. Choosing then the smallest possible gap that is still compatible with technological constraints, and selecting a polymer with a large linear electro-optic coefficient minimizes the

*π*-voltage

*U*, Eq. (1).

_{π}*f*

_{3 dB}=78GHz for a drive voltage amplitude as small as

*Û*=

*U*/4=1V. For explaining our PC waveguide design decisions, the following two subsections describe first a slow-wave PC slot waveguide PM and its properties, and then discuss the optimized PC slot waveguide design for a flattened dispersion curve.

_{π}### 4.1. PC slot waveguide

*W*

_{1}chosen to correspond to a W1.4 LD WG, the associated band diagram is displayed in Fig. 3. The low group velocity region is marked by an oval to the right. The figure is calculated with the guided-mode expansion (GME) method [15

15. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B **73**, 235114 (2006). [CrossRef]

*c*in the region below the light line. However, a high chromatic dispersion of

*C*>5ps/(mmnm) is also observed.

### 4.2. PC slot waveguide with dispersion flattening

*r*

_{1,2,3}and the distances between the hole centers

*W*

_{1,2,3}, see Fig. 4(a). As a result of the optimization process we find a set of W1.25 WGs that all provide a low group velocity over a wide spectral range. The frequency dependence of the resulting group velocity with various radii

*r*

_{2}as parameters is shown in Fig. 4(b). An increase of the parameter

*r*

_{2}decreases the group velocity, while a flat dispersion is maintained. For the value

*r*

_{2}/

*a*=0.36, the group velocity is 4% of the vacuum speed of light

*c*over a bandwidth of about 1THz. It is also possible to obtain a negative chromatic dispersion, which is for example the case for

*r*

_{2}/

*a*=0.30 or

*r*

_{2}/

*a*=0.38, Fig. 4(b). For all presented designs, the air hole diameters are larger than 200nm in order to meet fabrication constraints. The concept of the PC-WG with broadband low group velocity was experimentally verified previously [13

13. J.-M. Brosi, J. Leuthold, and W. Freude, “Microwave-frequency experiments validate optical simulation tools and demonstrate novel dispersion-tailored photonic crystal waveguides,” J. Lightwave Technol. **25**, 2502–2510 (2007). [CrossRef]

## 5. Modulator performance parameters

*f*

_{3 dB}and the phase modulator

*π*-voltage

*U*. The two parameters are derived and discussed in this section.

_{π}### 5.1. Modulation bandwidth of Mach-Zehnder modulator

*RC*limitations do not play a role for the presented structures as has been shown in the Appendix. The walk-off limited bandwidth depends on the termination of the CPWG, and here we discuss the two cases of a matched load and an open.

#### 1) Walk-off bandwidth, CPWG with matched load

*z*is not reflected at the end of the CPWG and maintains a spatially constant amplitude |

*U*|. Electrical and optical waves propagate in the same direction, but in general at different group velocities

*v*

_{g, el}and

*v*

_{g, opt}(group delays

*t*

_{g, el}and

*t*

_{g, opt}over the PM section length

*L*). The nonlinear interaction is maximum for co-directionally traveling waves (TW) with

*t*

_{g, el}=

*t*

_{g, opt}, and it decreases strongly if

*t*

_{g, el}≠

*t*

_{g, opt}. When the electrical and optical signal envelopes have acquired a phase difference of

*π*, the limiting modulating frequency

*f*

_{3 dB}is reached with ω

^{(TW)}

_{3 dB}|

*t*

_{g, opt}-

*t*

_{g, el}|=

*π*, i. e.,

16. C. Koos, P. Vorreau, P. Dumon, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “Highly-nonlinear silicon photonic slot waveguide,” in Technical Digest of 2008 Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, San Diego (CA), USA, Feb. 24–28, 2008, postdeadline paper PDP 25.

### 2) Walk-off bandwidth, CPWG with open-circuit

*ω*

_{3 dB}|

*t*

_{g, opt}+

*t*

_{g, el}|=

*π*,

16. C. Koos, P. Vorreau, P. Dumon, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “Highly-nonlinear silicon photonic slot waveguide,” in Technical Digest of 2008 Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, San Diego (CA), USA, Feb. 24–28, 2008, postdeadline paper PDP 25.

*v*

_{g, opt}≪

*v*

_{g, el}holds, then

*f*

_{3 dB}≈

*f*

^{(TW)}

_{3 dB}and electrically short MZM designs with and without matching load for the CWPG become nearly equivalent, resulting in Eq. (1).

## 5.2. π-voltage U_{π} of phase modulator

*π*-Voltage

*U*of its phase modulators. For a given PM length

_{π}*L*the voltage |

*U*| needed for a

*π*-phase shift is defined to be

*U*. For large modulation sensitivity,

_{π}*U*should be small. An optical wave propagating through a PM experiences a nonlinear refractive index change Δ

_{π}*n*in proportion to the electric modulating field

*E*

_{el}inside the nonlinear PM material,

*n*

_{poly}represents the effective linear part of the refractive index in the PM section, and

*r*

_{33}is the (scalar) linear electro-optic coefficient. The total phase shift of the optical wave due to the index change in a PM section of length

*L*is (following the formalism in Appendix Section A and in [16

16. C. Koos, P. Vorreau, P. Dumon, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “Highly-nonlinear silicon photonic slot waveguide,” in Technical Digest of 2008 Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, San Diego (CA), USA, Feb. 24–28, 2008, postdeadline paper PDP 25.

*k*

_{0}=2

*πf*

_{0}/

*c*is the vacuum wave number. In the process of deriving Eq. (5) the so-called field interaction factor Γ was introduced. It quantifies the strength of the nonlinear electro-optic interaction of modulating field and optical mode in a cross-section

*A*along a lattice period

*a*(see Appendix Eqs. (8)–(13) and [16

## 6. Optimized Mach-Zehnder modulator

*f*

_{3 dB}, Eq. (3), of the MZM amplitude modulator and for minimizing its

*π*-voltage

*U*, Eq. (7), the optical group velocity

_{π}*v*

_{g, opt}of the PC line defect WG, its length

*L*, the electro-optic coefficient

*r*

_{33}of the polymer, and the gap width

*W*

_{gap}need to be adjusted properly. For an integratable MZM with small length

*L*, the design bandwidth

*f*

_{3 dB}fixes the ratio

*v*

_{g, opt}/

*L*. Reducing

*v*

_{g, opt}then provides a small length

*L*. It also needs to be considered that with lower

*v*

_{g, opt}, the disorder-induced losses of the PC-WG increase [17

17. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. **94**, 033903 (2005). [CrossRef] [PubMed]

*v*

_{g, opt}cannot be made arbitrarily small.

*U*, the electro-optic polymer is chosen to have a large linear electro-optic coefficient of

_{π}*r*

_{33}=80pm/V [6

6. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express **13**, 5216–5226 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14-5216. [CrossRef] [PubMed]

*W*

_{gap}is chosen as small as compatible with the fabrication process. A gap width of

*W*

_{gap}=150nm can be fabricated to good accuracy with advanced lithographic processes; hence this width is fixed for the present design. Given the gap width

*W*

_{gap}, the maximum modulation voltage amplitude is limited in practice by the microwave source and by dielectric breakdown in the gap.

*v*

_{g, opt}, field interaction factor Γ, modulator length

*L*, and modulation bandwidth

*f*

_{3 dB}for various PC slot waveguide modulators without (W1.4) and with dispersion flattening (W1.25), as already presented in Section 4. The

*π*-voltage was kept fixed to

*U*=4V in all cases, which means that the modulation voltage amplitude

_{π}*Û*=

*U*/4=1V remained constant by adjusting the length

_{π}*L*according to Eq. (7).

*v*

_{g, opt}and Γ are calculated from simulations with the FIT method. As expected, the field interaction factor Γ increases and the modulator length

*L*decreases when lowering the group velocity

*v*

_{g, opt}according to Eqs. (6), (7). The bandwidth

*f*

_{3 dB}is calculated from Eq. (3) using the more exact numerical factor of 0.556 instead of 0.5. For a constant

*U*the estimate Eq. (1) would predict a constant modulation bandwidth

_{π}*f*

_{3 dB}, however, it shows a weak (1 : 1.4) dependence on

*v*

_{g, opt}(1 : 3.4). This is explained after Eq. (6): The field interaction factor Γ is only approximately proportional to 1/

*v*

_{g, opt}, and therefore

*U*∝

_{π}*v*

_{g, opt}in Eqs. (7), (1) is an approximation, too. For the dispersion flattened structures,

*f*

_{3 dB}is lower and shows a stronger dependence on

*v*

_{g, opt}compared to the structure with high dispersion.

18. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express **14**, 9444–9450 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-20-9444. [CrossRef] [PubMed]

^{18}cm

^{-3}) leads to additional optical losses of only about 1dB/mm [7

7. G. Wang, T. Baehr-Jones, M. Hochberg, and A. Scherer, “Design and fabrication of segmented, slotted waveguides for electro-optic modulation,” Appl. Phys. Lett. **91**, 143109 (2007). [CrossRef]

^{16}cm

^{-3}, optical losses will be mainly caused by disorder. With small device lengths the additional loss is expected to be tolerable.

*f*

_{3 dB}=78GHz at a length of only

*L*=80

*µ*m and for a small 1V drive voltage amplitude, we regard this to be an optimum structure for the discussed technological constraints.

## 7. Slow-light coupling structure

19. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express **11**, 2927–2939 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927. [CrossRef] [PubMed]

^{1}the strip WG mode into a slot-WG mode, Fig. 5(a). The design takes into account that the silicon on both sides of the gap needs to be electrically isolated as it is conductive and carries the modulation voltage, Fig. 2(a). FIT simulations of the strip-to-slot WG transition having a length of 7

*µ*m predict a transmission loss lower than 0.3dB and a reflection lower than -28dB.

*µ*m) to gradually slow down the PC WG mode. The calculated transmission and reflection curves are displayed in Fig. 6. The simulated structure comprises both the transition from a slot WG to a slow-light PC WG, Fig. 5(b), and the transition back to a slot WG.

## 8. Conclusion

*µ*m at a drive voltage amplitude of 1V. This allows transmission at 100Gbit/s.

## Appendix

## A. Field interaction factor Γ

*ε*on the permittivity of the material in Maxwell’s equations

*z*-direction, a complex envelope function

*A*is defined,

*f*

_{0}=

*ω*

_{0}/(2

*π*) and is assumed to be constant in the optical signal bandwidth centered at

*f*

_{0}. For the propagation constant

*β*of the mode we assume

*β*(

*ω*)=

*β*

_{0}+(

*ω*-

*ω*

_{0})

*v*

^{-1}

_{g, opt}neglecting chromatic dispersion. The orthogonality relation for twoWG modes

**Ĥ**

_{p, q},

**Ê**

_{p, q}subscripted with p and q, respectively, and having the same frequency can be derived following [20

20. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express **15**, 11042–11060 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-18-11042. [CrossRef] [PubMed]

*δ*

_{pq}is the Kronecker delta,

*P*

_{p}is the cross-section power of mode p, and the integration is performed over a cross-section perpendicular to the propagation direction. If Eqs. (10), (11) are substituted in Eqs. (8), (9), and both the mode orthogonality condition Eq. (12) and the slowly varying envelope approximation are used, a nonlinear propagation equation is obtained,

*A*=

*A*(

*z*,

*t*),

*U*=

*U*(

*z*,

*t*), Φ=Φ(

*z*,

*t*) and ΔΦ=ΔΦ(

*z*,

*t*) hold. The field interaction factor Γ is defined in Eq. (6),

*U*=

*U*(

*z*,

*t*) is the voltage across the gap. It is generated by the microwave field that causes an electro-optic refractive index change Δ

*n*=-

*r*

_{33}

*n*

^{3}

_{poly}

*U*/

*W*

_{gap}and a resulting phase change ΔΦ.

## B. Relation between field interaction factor Γ and group velocity *v*_{g,opt}

*ε*≪

*ε*, we substitute the modal fields Eqs. (10), (11) with

*A*=1 into Eq. (11) and obtain

*β*

_{0}and

*ω*

_{0}and find

**Ê**and the scalar product of Eq. (15) with

**Ê***. Then we add both relations, observe (

**e**

_{z}×

**Ĥ***)·

**Ê**=(

**Ĥ***×

**Ê**)·

**e**

_{z}=-(

**Ê**×

**Ĥ***)·

**e**

_{z}, and integrate over a volume

*v*=

*aA*with differential

*dV*=

*a*d

*A*, where d

*A*is the differential of the cross-section area

*A*in a plane

*z*=const. For PC WGs, the length a along the waveguide’s

*z*-axis corresponds to the size of the unit cell. We find

*f*

_{0}inside the volume under consideration. The integrand on the left-hand side of Eq. (16) is the real part of the

*z*-component of the complex Poynting vector

*A*is independent of the

*z*-position. So the volume integral may be written as an integral over the cross-section multiplied by the length a of the volume along

*z*. The result is

## C. Modulator bandwidth limitations by RC-effects

*RC*-effects. We discuss two different mechanisms that reduce the electrical voltage drop across the gap and lead to an associated 3dB bandwidth. These are the generator-determined bandwidth and the parallel-loss determined bandwidth. We find that these bandwidth limitations do not play a significant role for the presented structures.

*h*=220nm, PM section length and gap width

*L*=80

*µ*m and

*W*

_{gap}=150nm, linear refractive index of the PM section

*n*

_{poly}=1.6 (assumed to be the same at optical and electrical frequencies), width of the doped silicon slabs

*w*=3

*µ*m, vacuum dielectric constant

*ε*

_{0}, vacuum speed of light

*c*.

### 1) Generator-determined bandwidth

*Z*

_{L}=50Ω be terminated with an open, Fig. 7(a). We assume the PM section to be short compared to the modulation signal wavelength, so that the CPWG may be represented by a lumped gap capacitance

*C*

_{gap}=

*ε*

_{0}

*n*

^{2}

_{poly}

*hL*/

*W*

_{gap}for each PM. The voltage amplitude

*U*

_{gap}across the non-conductive gap

*W*

_{gap}(i. e., across the lumped capacitor 2

*C*

_{gap}) decreases with modulation frequency

*f*because of the generator impedance

*R*

_{G}=50Ω,

*U*

_{gap}| drops to |

*U*

_{G}|/√2. Assuming the previously assigned data and neglecting electric fringing fields, the total gap capacitance amounts to 2

*C*

_{gap}=5fF, and

## 2) Parallel-loss determined bandwidth

*Z*

_{L}, Fig. 7(b). Then, a modulating electrical wave traveling in

*z*-direction has a spatially constant amplitude |

*U*|. The resulting voltage amplitude |

*U*

_{gap}| across the non-conductive gap

*W*

_{gap}(PM capacitance per length

*C*′=

*ε*

_{0}

*n*

^{2}

_{poly}

*h*/

*W*

_{gap}) is reduced because of the finite resistivity of the doped silicon. For silicon sections having a width

*w*and a filling factor

*F*taking the reduction of the effective conductance by the air holes into account (conductance per length (

*R*′

^{-1}=

*σFh*/

*w*), we obtain

*f*

^{(R′C′)}

_{3 dB}denotes where the gap voltage |

*U*

_{gap}| drops to |

*U*|/√2. Assuming a conductivity σ=10Ω

^{-1}cm

^{-1}of the doped (

*n*

_{D}≈2×10

^{16}cm

^{-3}) silicon material and a filling factor

*F*=0.67, a bandwidth of

*f*

^{(R′C′)}

_{3 dB}≈120GHz results. In view of the envisaged MZM bandwidth of 78GHz, the parallel-loss determined limitation does not play a significant role either.

### 3) Other effects

^{7}cm/s. Over the 3

*µ*m witdh of the silicon slab, this would lead to a response time in the order of

*τ*

_{D}=50ps, resulting in a bandwidth in the order of 3GHz only.

*τ*

_{R}, inside which any charge perturbation in the doped silicon slabs (induced by the modulating field) is screened by a shift of the whole carrier ensemble. The dielectric relaxation time

*τ*

_{R}=

*ε*

_{0}

*ε*

_{r}/σ depends on the material’s conductivity σ and permittivity

*ε*

_{0}

*ε*

_{r}. For σ=10Ω

^{-1}cm

^{-1}as assumed above and with

*ε*

_{r}=12, a value of

*τ*

_{R}=0.1ps is obtained.

*τ*

^{-1}

_{res}=

*τ*

^{-1}

_{D}+

*τ*

^{-1}

_{R}≈

*τ*

^{-1}

_{R}, the

*faster*of the two effects is relevant, and the bandwidth limited by the dielectric relaxation time would be 1.6THz. As a consequence, carrier transit times do not limit the modulator’s performance.

## Acknowledgment

## Footnotes

1 | We thank an anonymous referee for pointing out a patent application, which describes such a structure: M. Lipson, C. A. Barrios, V. R. Almeida, R. R. Panepucci, and Q. Xu, United States Patent Application 20060228074. |

## References and links

1. | L. Liao, A. Liu, D. Rubin, J. Basak, Y. Chetrit, H. Nguyen, R. Cohen, N. Izhaky, and M. Paniccia, “40Gbit/s silicon optical modulator for high-speed applications,” Electron. Lett. |

2. | B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, “Electrooptic polymer ring resonator modulation up to 165GHz,” IEEE J. Sel. Top. Quantum Electron. |

3. | D. Rezzonico, M. Jazbinsek, A. Guarino, O.-P. Kwon, and P. Günter, “Electro-optic Charon polymeric microring modulators,” Opt. Express |

4. | Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, J. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, “Hybrid polymer/sol-gel waveguide modulators with exceptionally large electro-optic coefficients,” Nature Photonics |

5. | E. M. McKenna, A. S. Lin, A. R. Mickelson, R. Dinu, and D. Jin, “Comparison of |

6. | T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express |

7. | G. Wang, T. Baehr-Jones, M. Hochberg, and A. Scherer, “Design and fabrication of segmented, slotted waveguides for electro-optic modulation,” Appl. Phys. Lett. |

8. | Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature (London) |

9. | B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, “Compact electro-optic modulator on silicon-on-insulator substrates using cavities with ultra-small modal volumes,” Opt. Express |

10. | K. K. McLauchlan and S. T. Dunham, “Analysis of a compact modulator incorporating a hybrid silicon/electro-optic polymer waveguide,” IEEE J. Sel. Top. Quantum Electron. |

11. | L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett. |

12. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

13. | J.-M. Brosi, J. Leuthold, and W. Freude, “Microwave-frequency experiments validate optical simulation tools and demonstrate novel dispersion-tailored photonic crystal waveguides,” J. Lightwave Technol. |

14. | V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

15. | L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B |

16. | C. Koos, P. Vorreau, P. Dumon, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “Highly-nonlinear silicon photonic slot waveguide,” in Technical Digest of 2008 Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, San Diego (CA), USA, Feb. 24–28, 2008, postdeadline paper PDP 25. |

17. | S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. |

18. | L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express |

19. | S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express |

20. | G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express |

**OCIS Codes**

(130.0250) Integrated optics : Optoelectronics

(130.3120) Integrated optics : Integrated optics devices

(130.5296) Integrated optics : Photonic crystal waveguides

(130.4110) Integrated optics : Modulators

(130.5460) Integrated optics : Polymer waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 4, 2008

Revised Manuscript: March 10, 2008

Manuscript Accepted: March 10, 2008

Published: March 12, 2008

**Citation**

Jan-Michael Brosi, Christian Koos, Lucio C. Andreani, Michael Waldow, Juerg Leuthold, and Wolfgang Freude, "High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide," Opt. Express **16**, 4177-4191 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4177

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### References

- L. Liao, A. Liu, D. Rubin, J. Basak, Y. Chetrit, H. Nguyen, R. Cohen, N. Izhaky, and M. Paniccia, "40Gbit/s silicon optical modulator for high-speed applications," Electron. Lett. 43, 20072253 (2007). [CrossRef]
- B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165GHz," IEEE J. Sel. Top. Quantum Electron. 13,104-110 (2007). [CrossRef]
- D. Rezzonico, M. Jazbinsek, A. Guarino, O.-P. Kwon, P. Günter, "Electro-optic Charon polymeric microring modulators," Opt. Express 16, 613-627 (2008) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-2-613. [CrossRef] [PubMed]
- Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, J. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, "Hybrid polymer/sol-gel waveguide modulators with exceptionally large electro-optic coefficients," Nature Photonics 1, 180-185 (2007). [CrossRef]
- E. M. McKenna, A. S. Lin, A. R. Mickelson, R. Dinu, and D. Jin, "Comparison of r33 values for AJ404 films prepared with parallel plate and corona poling," J. Opt. Soc. Am. B 24, 2888-2892 (2007). [CrossRef]
- T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, "Optical modulation and detection in slotted silicon waveguides," Opt. Express 13, 5216-5226 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14-5216. [CrossRef] [PubMed]
- G. Wang, T. Baehr-Jones, M. Hochberg, and A. Scherer, "Design and fabrication of segmented, slotted waveguides for electro-optic modulation," Appl. Phys. Lett. 91, 143109 (2007). [CrossRef]
- Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature (London) 435, 325-327 (2005). [CrossRef] [PubMed]
- B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, "Compact electro-optic modulator on silicon-oninsulator substrates using cavities with ultra-small modal volumes," Opt. Express 15, 3140-3148 (2007) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3140. [CrossRef] [PubMed]
- K. K. McLauchlan and S. T. Dunham, "Analysis of a compact modulator incorporating a hybrid silicon/electrooptic polymer waveguide," IEEE J. Sel. Top. Quantum Electron. 12, 1455-1460 (2006). [CrossRef]
- L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, "High speed silicon photonic crystal waveguide modulator for low voltage operation," Appl. Phys. Lett. 90, 071105 (2007). [CrossRef]
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
- J.-M. Brosi, J. Leuthold, and W. Freude, "Microwave-frequency experiments validate optical simulation tools and demonstrate novel dispersion-tailored photonic crystal waveguides," J. Lightwave Technol. 25, 2502-2510 (2007). [CrossRef]
- V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Opt. Lett. 29, 1209-1211 (2004). [CrossRef] [PubMed]
- L. C. Andreani and D. Gerace, "Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method," Phys. Rev. B 73, 235114 (2006). [CrossRef]
- C. Koos, P. Vorreau, P. Dumon, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, "Highly-nonlinear silicon photonic slot waveguide," in Technical Digest of 2008 Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, San Diego (CA), USA, Feb. 24-28, 2008, postdeadline paper PDP 25.
- S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]
- L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Express 14, 9444-9450 (2006) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-20-9444. [CrossRef] [PubMed]
- S. J. McNab, N. Moll, and Y. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927. [CrossRef] [PubMed]
- G. Lecamp, J. P. Hugonin, and P. Lalanne, "Theoretical and computational concepts for periodic optical waveguides," Opt. Express 15, 11042-11060 (2007) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-18-11042. [CrossRef] [PubMed]

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