1. Introduction

Silicon-based photonic integrated circuits (PICs) are an attractive solution for next generation high-speed data transmission [1, 2]. Usually these circuits are realized using standard complementary metal-oxide semiconductor (CMOS) fabrication technology [3, 4] where silicon photonics offer low cost, low power consumption, and high bandwidth [2]. Recently silicon PICs have been demonstrated for both intra- or inter chip -link, and long haul telecommunication links [2]. Optical interconnects have recently emerged as promising solution for alleviating the bandwidth in modern computing electronics [5]. As a key part of optical interconnect on silicon chips, electro-optic (EO) modulators have been made great progress in recent years and attracted many interests [6]. In addition to that, optical networks have been growing up and integrated components such as optical modulators are widely developed and employed as key components for such application. Especially, the next generation optical networks are demanded to have the features of high capacity, high-speed and high- agility which require the aggregate carrier data per optical fiber to be extended toward 100 Tb/s [7]. This requires high performance modulators capable to deal with ultrahigh speed data rate of single-channel optical signals (100 Gb/s and higher in the future compared with current 10 and 40 Gb/s) [7, 8].

Conventional CMOS-compatible silicon optical modulator are usually realized using the linear EO (Pockels) effect rather than carrier depletion effect in order to achieve high speed operation [9-11]. Unfortunately, these modulators suffer from two main limitations

(i) CMOS compatibility sets limit to the required operating voltages [6]. Silicon is characterized by relatively low linear EO coefficients and hence requires relatively high modulating voltages. This problem has been solved by incorporating polymer with higher EO coefficient in the active region of the modulator leading to silicon-polymer hybrid modulators [12-15].

(ii) Advances in CMOS technology enable the fabrication of integrated electronics on nano-scale size. In contrast, the size of a conventional optical modulator is in order of operating wavelength or higher. A technology emerging recently which can scale down the dimensions of optical devices far beyond the diffraction limit is plasmonics [16]. This technology is based on surface plasmon polariton (SPP) which travels through a form of hybrid electrical/optical propagation [17-19]. A plasmon is a quasi-particle formed from the coupling of a photon and a travelling electron density wave at the interface between a metal and a dielectric [20]. Recently, plasmonic technology has been used to realize ultra-compact modulators suitable for integration with electronic components and hence meet footprint and efficiency requirement of PICs [21].

Several ultra-compact plasmonic modulators have been proposed in the literature for 50 Gb/s (and above) operation using silicon-metal [6, 22, 23] or silicon-polymer-metal hybrid structures [11, 24, 25]. The proposed modulators have been designed using Mach-Zehnder interference (MZI) [10, 22, 25-30], nonresonant (forward) waveguide [9, 15, 31-34], or microring resonator (MRR)-based structure [35-37]. Plasmonic modulator for high-level modulation have been also proposed using a combination of MZI and waveguide topologies [11-13]. The microring resonator attracts increasing interest as a promising photonic device for integrated optical interconnects and novel photonic architectures which can be implemented easily in silicon photonic platforms. The main advantages of this device are

(i) It can be configured either as an optical modulator or a wavelength division multiplexing drop filter [38].

(ii) It offers smaller footprint than the Mach-Zehnder modulator (MZM) [16].

(iii) When fabricated using silicon-on-insulator (SOI) platforms, the microning resonator exhibits a great potential to build space, power, and spectrally efficient on-chipphotonic networks that could be seamlessly integrated with CMOS electronics [39].

(iv) The microning modulator exhibits smaller chip size and capacitance, resonant-enhanced efficiency, and high wavelength selectively in comparison to MZM [39].

(v) When compared to MZM, ring resonator has the potential for low power consumption and array integration [40].

Unfortunately, the performance of MRRs is very sensitive to the operating wavelength, due to their high thermal sensitivity, and to the geometric and material parameter variations, due to their small size. Therefore, the design of microning modulators should be addressed carefully.

Recently, Xu et al. [36, 37] have proposed a race track ring-based optical modulator employing an EO polymer infiltrated silicon- plasmonic hybrid phase shifter. Their simulation results reveal a 15dB extinction ratio under 1.2V.

This paper presents comprehensive analysis and detailed performance investigation for this modulator structure when it is designed to operate at optical telecommunication wavelengths. The results can be used as a guideline to design ultra-compact and high speed plasmonic modulator suitable for implementation in CMOS platform. The first part of this paper constructs the theoretical framework required to design the modulator and develops general expressions to assess its performance. Simulation results related to modulator performance and its capability to be implemented in high bit rate optical communication systems are given in part II.

The simulation results stand heavily on COMSOL 4.3b and partially on OptiSystem 12 software packages.

2. Device Description and Design Methodology

The silicon-plasmonic racetrack modulator proposed in [36, 37] will be investigated in this paper. This modulator consists of racetrack resonator coupled to a bus waveguide as shown in Fig. 1. The plasmonic slot waveguide with EO organic cladding material is positioned above one of the straight arms (silicon waveguide) of the resonator. This principle is similar to directional coupling between two waveguides. The plasmonic photonic hybrid directional coupler consists of two arms. The upper arm is silver-polymer-silver plasmonic slot waveguide while the lower arm is silicon dielectric waveguide embedded in silicon di-oxide. Different widths of silicon waveguide are obtained according to phase matching condition corresponding to different polymer slot widths.

Figure 1. Structure of Silicon-plasmonic racetrackresonator modulator

Two guided eigen modes (quasi-even and quasi-odd modes) are formed consequently in the overlapping area between the two the photonic and plasmonic waveguides. The propagation constants of eigen modes being different, the power exchange between the waveguides can be represented as a beating between these two modes. The basic principle of the hybrid coupler is illustrated in Fig. 2.

Figure 2. Basic principle of plasmonic photonic hybrid coupler

The hybrid directional coupler can act as a phase shifter by applying static electric field via Pockels effect, where is the modulating voltage and is polymer slot width. The refractive index of the polymer is changed by , where is EO coefficient, is the polymer refractive index, and is the optical confinement factor of the slot. By modulating the refractive index of the polymer in the slot, the information is encoded in the phase of the SPP.

To design the hybrid phase shifter, some important factors should be taken into account.

(i) Phase matching condition: The real part of the effective refractive index of the fundamental photonic mode should be equal the real part of the effective refractive index of the fundamental plasmonic mode. This condition is satisfied for a specific polymer slot width as represented by the operating point A in Fig. 3. Thus the required silicon waveguide width is obtained. Note that the effective index of the plasmonic mode is independent of .

Figure 3. Real parts of the effective refractive indices of photonic and plasmonic modes as a function of silicon waveguide width

(ii) The length of the phase shifter should be equal even number of coupling length to ensure maximum transfer of power from the plasmonic waveguide to the optical waveguide at the end of the phase shifter. Coupling length is a measure of beating length of quasi-even and quasi-odd modes inside the coupler and can be related to propagation constants of the two eigen modes by . Here and represent propagation constants for the quasi-even and quasi-odd modes, respectively.

3. Coupling between Silicon and Plasmonic Waveguides

The coupling between the modes of plasmonic waveguide and silicon waveguide can be investigated using the concept of parallel two-waveguide coupler as shown in Fig. 4. Both waveguides are assumed to be homogeneous along the direction of propagation (i.e., z axis) with refractive index distribution which is in general a function of x and y axes.

Figure 4. (a) Parallel two-waveguide coupler model to characterize the interaction between plasmonic and silicon waveguides. (b) Refractive index distribution in the transverse plane of the coupler

The refractive index distribution in the three layers of Fig. 4a can be expressed as

(1)

When the two parallel waveguides are far apart, the electric field modes of propagation of the individual waveguides can be expressed as

(2a)

(2b)

When the two waveguides are separated by a short distance, the coupling effect should be taken into account. In this case, the electric field of the wave propagating in the coupled-waveguide structure can be expressed as [41]

(3)

where the mode amplitudes and are introduced to take into account the coupling effects along the axial direction of the waveguide.

The electric field of the propagating wave in the coupler strucure should obey the scalar wave equation

(4)

where is the electrical field complex amplitude, is the Laplacian operator, c is the speed of light in vacuum, and

(5a)

with

(5b)

(5c)

Note that the two functions and obey the orthogonality since they refer to separate transverse regions

(6)

where

The transverse behavior of the field in a waveguide can be described by using the wave equation [42, 43]

(7)

where is the transversal Laplacian operator. Therefore, the individual coupler waveguides modes satisfy these equations

(8a)

(8b)

Subsituting eqn. (3) into eqn. (4) and using the assumption of slow variation of mode amplitude over z yeilds

(9)

In writing eqn. (9), eqns. (8a) and (8b) are used under the assumption that the two coupler waveguides are not too close such that

(10)

Taking the scalar product of eqn. (9) with and , respectively, and integrating over the entire xy plane yields two-coupled mode equations

(11a)

(11b)

where

(12a)

and

(12b)

In eqns. (12), represents the permitivity of vacuum.

The coupling constants and represent only a small correction to the propagation constants and , respectively. The terms with and in eqn. (11) result from the dielectric perturbations to one of the waveguide dueto the presence of the other waveguide. Under the assumption that the individual waveguides modes are locallized almost totally in its waveguide, then according to eqn. (12b) one can set These leaves eqns. (11a) and (11b) with two coupling constants and which form a complex conjugate . Under this assumption, eqns. (11a) and (11b) can be rewritten as

(13a)

(13b)

where is assumed to be real without loss of generality and set equals to κ.

Then

(14a)

In similar way, one can arrive to the following equation starting from eqn. (13b)

(14b)

A general solution of eqns. (14a) and (14b) takes the form

(15a)

(15b)

where q is a complex propagation constant.

Subsituting eqns. (15a) and (15b) into eqns. (14a) yeilds

Then

(16a)

(16b)

Similarly, subsituting eqns. (15a) and (15b) into eqn. (14b) yeilds

(16c)

(16d)

The relations in eqns. (16) are satisfied with nonzero values of and if the possible values of q obey the following dispersion relation (obtained by setting the determinant of the cofficient matrix to be zero)

(17)

Let eqns. (16) be solved under the following boundary conditions

Dividing eqn. (16a) by (16b) yields

This gives

(18a)

(18b)

(18c)

Using eqn. (15b) with and recall that give

(19a)

Using eqns. (18a) and (18b) into eqn. (15a) and recall that yeilds

(19b)

4. Design Concepts of the Plasmonic Ring Modulator

Based on Fig. 5, the linear input-output relationship of the bus-ring coupler can be described by the matrix relation

(20)

where

Straight-through coupling coefficient between bus and ring waveguide.

Cross-coupling coefficient between bus waveguide and ring waveguide.

Figure 5. Model of single racetrack resonator with hybrid phase shifter and one bus waveguide

The superscript * denotes the conjugated complex.

The round trip in the ring is given by

(21)

where

From eqn. (21)

(22)

Substituting eqn. (20) into eqn. (22) yields

(23a)

From eqn. (20)

(23b)

The transmission function of the ring modulator can be expressed as

(24)

The transmission function of the hybrid phase shifter can be obtained from

(25)

where is the electric field concentrated on silicon waveguide consisting the hybrid phase shifter. By using coupled mode theory between plasmonic waveguide and silicon waveguide and setting the plasmonic feild yields

(26)

where is the mean attenuation length of the hybrid waveguide . Here is the imaginary part of the effective refractive index of the plasmonic mode .

The transmission function of the hybrid phase shifter can be written as

(27a)

where

(27b)

In eqn. (27a), is the phase velocity mismatch between the silicon waveguide photonic mode and the metal-dielectric-metal (MDM) plasmonic mode [37]

(28a)

where is the propagation constant of the silicon waveguide mode whose effective index is assumed to be real (lossless waveguide). Further, is the real part of the complex propagation constant of the MDM plasmonic mode

(28b)

where is the complex effective index of the plasmonic mode.

In eqn. (27b), the parameter s is defined as [37]

(28c)

The coupling length .

By assuming that coupler's transmission and coupling coefficients are related by (i.e., lossless coupler), then eqn. (24) becomes

(29)

Substituting eqns. (27a) and (27b) into eqn. (29) yields

where which reflects the effective loss of the resonator.

(30)

Equation (30) can be rewritten as

(31)

where

(32a)

(32b)

(32c)

Equation (31) can be rewritten as

The power transfer function is given by

(33)

The minimum and maximum values of can be obtained by searching for the values of which leads to . The following equation is obtained under this condition (see eqn. (33)).

Values of that satisfy the solution of the above equation should make This occurs when

At resonance , eqn. (33) reduces to

(34)

where is evaluated at the resonance wavelength .

It is worth to mention here that according to eqn. (32c), the resonance condition implies that the following two subconditions should satisfied simultaneously.

(35a)

(35b)

The first subcondition is satisfied when where is a positive integer. Note that negative integer values for cannot be used since s and are positive. Recall that the coupling length . This implies that and hence is even number of the coupling length . In this case, the optical power coupled from the silicon waveguide to the MDM plasmonic waveguide equals to zero at (see eqn. (27)). This means that at the end of the phase shifter, the optical power is completely transfer from the hybrid waveguide to the silicon waveguide.

The second subcondition (eqn. 35b) can be used to estimate the resonance wavelength

(36a)

Thus the optical circumference of the ring is related to by

(36b)

Investigating eqn. (36b) reveals the following facts

(i) should be a positive integer to yield a positive value for L.

(ii) For a given resonance wavelength, higher coupling efficiency between the two waveguides (i.e., δ is smaller) leads to a slightly shorter ring circumference.

(iii) The resonance condition in conventional optical resonators (i.e., the round trip optical length should be even number of resonance wavelength) is not satisfied in the resonator under investigation unless strong coupling between the two waveguide exists .

Using eqn. (35a) into eqn. (32b) yield Hence eqn. (34) is simplified to

(37)

Note that tends to zero at a critical coupling coefficient

(38)

Note further that approaches 1 when (i.e., lossless modulator) as shown in Fig. 6.

Figure 6. Relationship between minimum transmission and effective loss of theresonator

5. Bandwidth of the Transmission Characteristics

The analysis is carried further to determine the 3-dB bandwidth of the power transmission characteristics . The maximum value of occurs when where is an odd integer

(39)

The low and high half-power points and satisfy the relation

After simplification, one gets

(40a)

Using the approximation gives

(40b)

Note that the solution of eqn. (40b), which is derived under the assumption of , yields . Therefore, the full width-at-half maximum (FWHM) bandwidth of the transmission spectrum .

The analysis of the transmission spectrum bandwidth is extended further to obtain a simplified expression related to practical designs. For high efficiency operation, the design of the ring modulator should satisfy the following two requirements

(i) High-quality factor (Q) resonator to reduce the required applied voltage to switch the device from the ON state to OFF state (or vice versa).

(ii) Operation under critical coupling to ensure maximum extinction ratio.

The first requirement implies that deviates slightly from the resonance condition where is a positive integer. Therefore, can be treated as a small angle relative to . Hence one can use the approximation to simplify eqn. (40b). The final result is

(41a)

where

(41b)

(41c)

From eqn. (32c)

(42)

where and can be considered as the resonance wavelength when . Here denote the lower and upper half-power wavelengths, respectively.

Using eqn. (42) into eqn. (41) yields

(43)

In deriving eqn. (43), the variation of the refractive index with wavelength is not taken into account and it is assumed that for high Q-resonator. Otherwise, should be replaced by the group refractive index .

It is worth to examine eqn. (43) under the condition of critical coupling where In this case

(44)

The resonance quality factor is given as

(45a)

(45b)

Equations (43) and (45a) can be rewritten in another forms after using (assuming strong coupling). The results are

(46a)

(46b)

Under critical coupling

(46c)

(46d)

Investigating eqns. (46) reveals that at fixed value of , both and are proportional to the resonator circumference. This leaves the quality factor to be independent of L. Note further that approaches when . This case corresponds to a lossless resonator where .

6. Chirp Characteristics

This section addresses the chirp (frequency modulation effect) when the device is used for analog intensity modulation. The phase characteristics of the output transfer function is governed by the following expression

(47)

The frequency chirp is related to by

(48a)

Since with is the speed of light in vacuum, the wavelength chirp is then computed as

(48b)

Note that is a function of the angle which is a function of the applied voltage.

From eqn. (32c), it is clear that is a function of and the coupling length is voltage dependent. Therefore, the frequency chirp can be calculated as

(49)

where is the total applied voltage consisting of a bias voltage and the ac modulating signal . Note that although the bias voltage does not affect it plays a key role in determining the chirp through the factor .

From eqn. (47) and after using the relation , one gets

(50)

where

(51a)

(51b)

(51c)

(51d)

The factor can be computed from

(52a)

From eqn. (32c)

(52b)

Few remarks related to the chirp of the modulator under investigation are given here

(i) To reduce the chirp effect when the modulator is used for intensity modulation, the operating point (i.e., the bias voltage) should be chosen carefully to achieve low value.

(ii) Positive DC voltage should be used to achieve low chirp intensity modulation.

7. Modulation Bandwidth

Three factors affect the modulation bandwidth of the modulator under investigation

(i) Modulator RC time constant

(ii) Modulator photon lifetime

(iii) Bandwidth limitation of the derive circuit.

Parallel-plate capacitance model can be used to estimate the modulator capacitance that seen between the two contacts

(53)

where the relative permittivity (dielectric constant) of the polymer is taken as . Here represents the permittivity of the vacuum. Further and represent slot height and width, respectively. The bandwidth due to RC time constant is

where is the loading resistance and sets to in the simulation. Note that is directly proportional to slot width and inversely proportional to both slot height and phase shifter length.

The modulation bandwidth due to photon lifetime is related to photon life time by [44]

(54)

The photon lifetime in optical resonator is related to its equality factor as follows. The quality factor is defined as [45]

(55)

The stored energy is lost at a rate of (per unit time) which is equivalent to the rate per cycle, where T is the period of the optical wave. Therefore

(56a)

Thus

(56b)

Using eqn. (43) into eqn. (56b) yields

(57a)

Under critical coupling

(57b)

Note that shorter L leads to higher .

8. Electrical Power Consumption

The electrical power dissipated during switching the modulator from one state (applied voltage = 0) to another state (applied voltage=) can be considered mainly due to charging (or discharging) the modulator capacitor. To calculate this dissipated power, the modulator under investigation is modeled simply as a series RC network driven by a voltage source through a switch S whose operating state (i.e., ON or OFF state) is controlled by the binary modulating bits (see Fig. 7).

Figure 7. Simplified model to calculate the electrical power consumption during the switching modulator state

When the switch is turned off by the applied bit, the capacitor starts to charge gradually through the resistance R. During the transient time, the capacitance voltage is given by

(58a)

and the current passing through the RC circuit is

(58b)

The average power dissipated during one bit time , where is the bit rate, can be calculated as

(59a)

For practical applications, the condition and therefore

(59b)

Equation (59b) indicates clearly that the dissipated power scales linearly with the bit rate and with square of the applied voltage. Note that is independent of the resistance R.

Few remarks related to the switching dissipated power can be noted here

(i) The power consumption associated with discharging the modulator capacitance (i.e., the switch is off in Fig. 7) is also governed by eqn. (59).

(ii) In driving the modulator with real binary digital signal, the number of charge discharge cycles depends on the pattern of bits. In average, there is one complete charge/discharge cycle every four bits under the assumptions of equal numbers of "Ones" and "Zeros" are transmitted [46]. This can be understood by observing two successive bits which fall into four equally likely sequences (11, 00, 01 and 10). In the first two sequence, there is no change of state, so no energy is dissipated in charging the capacitance [46]. Therefore, the actual average dissipated power is . In other words, the average dissipated energy per bit is given by

(60)

(iii) To reduce the switching dissipated energy, one should design the modulator with small capacitance . Note that the modulator capacitance depends only on the slot structure parameter (i.e., polymer refractive index and slot dimensions ).

(iv) If the stray capacitance due to the connection with driving circuit is included in the calculation, then the effective modulator capacitance increases and this will increase the dissipated power.

9. Extinction Ratio and Insertion Loss

Assume the modulator is excited by an optical signal having a wavelength equal to modulator resonance wavelength associated with zero applied voltage . (See Fig. 8a). Figure 8b shows the transmission characteristics as a function of wavelength for two values of applied voltages. The applied voltage shifts the resonance frequency. Positive values of increases the polymer refractive index and hence introduce red shift to the resonance frequency.

In decibel scale, the insertion loss (IL) and extinction ratio (ER) are defined as

(61)

(62)

Note that

Hence

(63)

where is the insertion loss when .

Note that the summation of insertion loss and extinction ratio is independent of applied voltage and it is equal to

(64)

Figure 8. (a) Basic modulator model. (b) Modulator transmission characteristics versus wavelength

10. Conclusions

A design methodology has been explained for an optical ring modulator incorporating silicon-polymer-metal hybrid plasmonic phase shifter. Analytical expressions have been derived to characterize the transmission characteristics and performance parameters of this modulator. The results can be used as a guide line to design compact modulators for optical communication wavelengths.

ACKNOWLEDGMENTS

The work is a part of a PhD research program in the Institute of Laser for Postgraduate Studies, University of Baghdad, Iraq. One of the authors, Mrs. Al-mfrji, wishes to thank the College of Engineering at Al-Nahrain University, Baghdad, Iraq, for offering her the PhD scholarship.