## Abstract

The free carrier absorption effect in silicon modulation is a detrimental behavior that can influence the crosstalk of interference-based optical switches. Based on the experimental analysis of a 2×2 *p-i-n* silicon switch, we give a conservative estimate of the crosstalk ability of Mach-Zehnder optical switches. Experimental result shows that, while using a 1475μm-long phase shifter, the loss penalty almost reaches 1.45dB/π, which deteriorates the most ideal crosstalk to just 30dB. The possible solutions to overcome this limitation are also discussed at the cost of the other device performance.

© 2009 Optical Society of America

## 1. Introduction

Ever since the emergence of integrated optics, photonic devices are continuously pursuing even “smaller, swifter, smarter, simpler and saving more energy and cost” (5S) implementations. In recent years, optics on silicon platform opens the door to resolve the current micro-electronics bottlenecks [1]. However, it still remains a challenge to externally modulate light effectively with high performances [2]. Due to the promise of high-speed modulation, free carrier dispersion (FCD) effect has gained widespread application in optical modulation. Meanwhile, Mach-Zehnder interference (MZI) structure is inherently considered as the mainstream way to implement modulators [3, 7–13], switches and filters. Its advantages in bandwidth and fabrication are irreplaceable by the resonant structures such as ring, disk, Bragg grating, even photonic crystal. Hence, it is a natural thing to combine the FCD with the MZI structure to implement broad-bandwidth silicon modulators or switches.

In the beginning, because of the relatively weak refractive index change (<10^{-3}) and the carrier’s lifetime in this effect [4], most work was devoted to enhance the modulation efficiency and improve the modulation speed. However, the influence of the coexisting free carrier absorption (FCA) was scarcely studied in a MZ-based device. As shown in Table.1, the extinction ratios (ER) are no more than 22dB even in the fabricated modulators even under the condition of wavelength scanning [11], mainly due to the voltage-controlled loss penalty to the phase modulation. Without this negative effect, the low-speed thermo-optical silicon switches can readily achieve a high switching crosstalk of 25dB beyond [5, 6].

The 1×2 or 2×2 optical switches are extremely crucial to construct a switching array with large port count [5]. As for a silicon MZ modulator, one can achieve infinite ER at the expense of the excess loss in theory, as long as the OFF state is assumed to be totally off. However, it is not the case for a 2×2 MZ switch, or with even larger port counts, whose crosstalk (CT) is quite limited by the FCA. Clearly, we should take good care of the every switching state simultaneously. This paper focuses on the influence of FCA to an MZI-based optical switch supported by theory and our recent experimental results.

## 2. Theoretical prediction

#### 2.1 Deterioration of power imbalance to the switching performance

The output intensity of an MZI device is determined by the loss and phase information of the beam from each path. It is well know that the power imbalance deteriorates the completeness of interference. Specifically, for the case of two-beam interference in a modulator, the most ideal (namely possible) extinction ratio (ER) can be expressed by

where *r* is the power imbalance factor closely before the interference occurs. According to this expression, Figure 1 schematically shows the relation between ER and *r*, which presents an intuitive estimation of the ER under the condition of unbalanced interference. It indicates that, in order to obtain a modulator with ER beyond 30 dB, *r* should be within the scope of [0.9, 1.1]. If the power imbalance problem is serious, typically with *r*<0.65, the most ideal crosstalk can never exceed 20dB. While adding a coupler to compose an 2×2 optical switch, both the two states should be taken good care of to achieve high performance. If the added coupler is ideally assumed to be lossless and free from the phase distortion, the switching crosstalk (CT) is identical to the ER, and can be calculated by Eq.(1) as well.

#### 2.2 Free carrier absorption (FCA) effect in silicon

It is necessary to examine quantitatively the free carrier absorption effect in silicon. The *Kramers-Kronig* analysis of optical absorption spectrum enables R. Soref *et.al* to investigate the FCD effects in silicon, which is formulated by the expressions below at a wavelength (*λ*) of 1.55μm [4]

where *ΔN* and *ΔP* are the concentration change of electron and hole in cm^{-3}, respectively. *Δn* and *Δα* are the refractive index and absorption coefficient variations. The subscripts “*e*” or “*h*” means the contribution is from electron or hole, respectively.

As indicated in the above expressions, the main problem in the FCD effect is the inherent coincidence of refractive index change and modulation loss penalty. According to the terms in Eqs.(2–3), The modulation efficiency of the electron and the hole is compared in Fig. 2(a). Clearly, the *hole* effect benefits larger refractive index change than the electron effect and less loss penalty in the typical injection concentration nowadays of ~10^{17-18.5}cm^{-3}. The critical point gets close to 10^{20} cm^{-3} for electron effect to exceed the *hole* effect. Hence, in order to relieve the influence of additional carrier absorption, it is strongly recommended to enable the *hole* effect to be as dominant as possible. Although a device injecting only holes might be difficult to implement, a reverse-biased diode might work for this [13]. Hence, in this paper, the cases *without* and *with* electron effect are treated respectively to discuss the performance limit of an MZI-based device.

The required length of a π-phase shifter can be estimated by *L*
_{π}=*λ*/(2∣*Δn*∣). By using Eqs.(2–3) and the expression

Figure 2(b) shows the loss penalties *l*
_{(dB/π)} for a phase shift of π at specific concentration changes with and without *ΔN*. By taking both the electron and hole into account, the typical values of *l*
_{(dB/π)} with the carrier densities varying from 10^{15}cm^{-3} to 10^{17-18.5}cm^{-3} are within the range from 0.7dB to 1.5dB. Even in the case only *hole* effect exists, the *l*
_{(dB/π)} value reaches 0.83dB (i.e. *r*=0.8265) with *ΔP* =5×10^{17}cm^{-3} [3].

#### 2.3 Phase shifter with FCA effect installed into an MZI-based switch

In Fig. 3, the phase shifters using the FCA effect are installed into an MZI-based switch device now. For the sake of description, the superscripts “*in*”, “*md*” and “*out*” mean the input, middle (positions after modulated) and output positions; “*a*”, “*ψ*” and “*φ*” are the amplitudes, phase variables and phase modulation magnitudes; the subscripts “*A*” and “*B*” mean the upper and lower paths, respectively. *T*
_{MMI} is the transfer matrix of a 2×2 MMI coupler.

As listed in Table 2, there are four typical schemes to realize 2×2 optical switching. The letter “ *t* ” is the loss penalty in a π-phase shifter and equals to 10^{- l;(dB/π)×π/10}; “MPI” stands for maximum power imbalance before interference on different states, i.e. max(*a*
_{A}
^{md}/*a*
_{B}
^{md}); “MMM” stands for maximum modulation magnitude on individual phase shifter. The possible CT of all the schemes can be obtained by replacing the variable *r* in Eq.(1) with MPI.

For the Case 1 in Table 2, where two interferometric arms are initially in phase (*ψ*
_{A}
^{in}-*ψ*
_{B}
^{in}=0) or out of phase (*ψ*
_{A}
^{in}-*ψ*
_{B}
^{in}=π) with equal intensities (*a*
_{A}
^{in}/*a*
_{B}
^{in}=1), the MPI equals to *t* and the CT can just reach 26.5dB for *ΔP*=5×10^{17}cm^{-3} and 21.5dB for *ΔP*=*ΔN*=5×10^{17}cm^{-3}at best. There is an approach to reduce the loss penalty by pre-biasing the initial power ratio (*a*
_{A}
^{in}/*a*
_{B}
^{in}) of the two arms to be *t*
^{-1/2} or *t*
^{1/2} (Case 2). This way makes the power imbalance factors *a*
_{A}
^{md}/*a*
_{B}
^{md} on one state to be *t*
^{1/2} and another 1:*t*
^{1/2}, respectively. The loss penalty is distributed to both the two states, which makes an optical switch with the CT of 32.5dB for *ΔP*=5×10^{17}cm^{-3} and 27.5dB for *ΔP*=*ΔN* =5×10^{17}cm^{-3}. As indicated in the Case 3, pre-biasing a constant phase shifter π/2 (*ψ*
_{A}
^{in}-*ψ*
_{B}
^{in}=π/2) is also an option to achieve the same CT as the Case 2, which is adopted in our experiment to reduce the maximum modulation magnitude (MMM) to π/2. Phase is also easier than initial power ratio to be biased by considering the practical realization. Study shows that these two biasing methods may achieve the best CT one can obtain. Since the MPI still takes the value of *t*
^{1/2}, the pull-push mechanism in the Case 4 cannot further improve the crosstalk any more. It just reduces the MMM to a quarter of that in the case 1.

Obviously, a higher injected concentration produces larger carrier concentration change, and thus reduced the device length. But, in the meanwhile, the loss penalty *l*
_{dB/π} increases dramatically, which further reduces the device CT seriously, as shown in Fig. 2(b). Hence, there is a tradeoff in carrier injection concentration between the length of phase shifter and the desired CT. In order to give an intuitive guideline for device optimization, the required lengths of π-phase shifter *L*
_{π} and the possible CT are simultaneously calculated to observe the tradeoff in Fig. 4 by varying the initial and final carrier densities.

Clearly, the CT and *L*
_{π} are just related to the absolute concentration changes. Thus, we can use a “Watershed” line to divide the picture into the upper part and the lower one for the cases without and with electron effect, respectively. The blue-to-red-colored contour and the black marks are for the CT (dB). The blue contour lines and marks are for the order of L_{π} (μm) (i.e. *log*
_{10}
*L*
_{π}). Basically, even without electron, it is impossible to obtain a 30dB-CT switch by using a π-phase shifter shorter than 10^{3}μm (see the marks “3”). If a π-phase shifter as compact as 2×10^{2}μm-length is needed, the crosstalk of the optical switch is no more than 25dB in theory. The case with both the electron and hole effects is even worse in the typical injection concentration. As what we predicted, this is a serious problem to realize switching array while involving the FCD effect.

## 3. Experimental demonstration and analysis

A 2×2 MZI switch fabricated by CMOS-compatible technique is taken as an example here. As shown in Fig. 5, this device consists of an MZI structure by cascading two 2×2 MMI coupler, and uses the conventional *p-i-n* structure. The wafer was (100) orientation with p-doping (14Ω.cm<*ρ*<22Ωcm). Figure 5 also presents the rib waveguide cross section, with 1μm SiO_{2} buried layer. The testing uses a single-wavelength light source at 1550nm. The transmission curves from each output port were measured by *lensed fiber-to-fiber* coupling at different injected currents. It should be noted that the positive and negative current means the cathode A works and cathode B works, respectively.

One can fit the transmission characteristic curves with the formula of the free carrier dispersion and the derived matrix multiplication expression below

All the parameters in the above expression are shown in Fig. 3. According to the experimental results and *p-i-n* theory, the fitting of the transmission characteristic curve can be done like
this. First, there is a quasi-linear relation *ΔN*
_{avr}+*ΔP*
_{avr}=*C _{INP}*×

*ΔI*between the electric current change

*ΔI*and the average carrier density change

*ΔN*

_{avr}+

*ΔP*

_{avr}in the space where the current passes through [14].

*C*is the proportional coefficient. Second, one should write down the electric currents of all the ON/OFF points. Between two neighborhood ON/OFF states,

_{INP}*ΔN*

_{avr}+

*ΔP*

_{avr}should produce a fixed refractive index change

*Δn*through Eq.(2) to produce an equal π-phase shift, which are related by π=

*Δn*

_{eff}(2π/

*λ*)

*L*

_{arm}=(

*ΓΔn*)(2π/

*λ*)

*L*

_{arm}.

*Γ*is a factor related to the overlapping of the carriers and optical fields.

*L*

_{arm}is the length of the phase shifter. Third, by using the least standard variance method, one can scan out the right CINP value numerically, which is correspondent to the minimum standard variance of

*Δn*between the neighborhood ON/OFF points. Now, the relation between the injected current and the carrier-induced phase/loss value are fitted.

In our device, *L*
_{arm}=1475μm; *C _{INP}* is about 2.1×10

^{16}cm

^{-3}/mA; the value of

*Γ*is estimated to be 0.34. The effective carrier injection density

*ΔN*

_{eff}=

*ΔP*

_{eff}is about 4.6×10

^{17}cm

^{-3}for 7π-phase shift, which is defined by

*Δn*

^{-1}(

*λ*/2/

*L*

_{arm}), where

*Δn*

^{-1}is the inverse function of Eq.(2). In Fig. 6, we plot the induced phase changes and the power losses during the fitting process. The loss penalty

*l*

_{dB/π}is about 1.45dB/π on account of both the hole and electron effects. This loss value increases dramatically by increasing the injected current.

To plot the fitting curves of the transmission spectrum, the pre-biased initial power ratio and the constant phase difference are intentionally guessed to align the experimental figure. In the device, *a*
_{A}
^{in}/*a*
_{B}
^{in}=1.012 with an shortened 3-dB MMI splitter; *ψ*
_{A}
^{in}-*ψ*
_{B}
^{in}=π/3.2 due to the fabrication error; (*a*
_{A}
^{in}
*a*
_{B}
^{in})^{1/2}=-3.6dB resulted from the excess loss. The transmission curves and the fitted ones are presented in Fig. 7. Because the initial state is not symmetrical as expected, the transmission spectrums are different for the cases the cathode A works (CT>19.5dB) and the cathode B works (CT<17.1dB). The envelope through the transmission valleys shows the modulation loss deteriorates the CT of the optical switch seriously. The power imbalance factors at the four “ON” and “OFF” states are 0.65, 0.93, 0.81(MPI) and 0.53, respectively. Even if this switch is symmetrical at initial state (i.e. *a*
_{A}
^{in}/*a*
_{B}
^{in}=1, *Δ*
_{A}
^{in}=*Δ*32^{in}) and the shallow valley of the transmission spectrum can be fine detected, the most ideal crosstalk one can obtain is no more than 30dB.

## 4. Discussions

Unlike the speed and modulation efficiency, the loss penalty cannot be eliminated by simply scaling down the device dimension to improve the efficiency in theory. Both the real and imaginary parts of the complex dielectric permittivity vary by increasing the injected density. Hence, it makes no sense to the loss penalty for unit phase change.

As Fig. 2(b), the loss penalty decreases with the decreased injection concentration, but not significantly. This point is indicated by our device in above section. What is worse, low-density injection makes the device length increasing exponentially (typically possible CT equals to 29.6dB and *L*π=4000μm while the effective carrier injection density *ΔP*=*ΔN*=5×1031cm^{-3}) and then unsuitable for large-scale integration.

One can use specific structures (e.g. ring-coupled MZI structure) to reduce the necessary phase shift and further favor the two switching states in power balance to improve the CT. But these methods are not only complex but also at the cost of the operation bandwidth. Another approach is to dynamically serve the pre-biased splitting ratio, which should maintain the imbalance factor as close to unity as possible for each switching states. The co-working of the extra electrodes greatly leads to the complexity of the device structure and controlling system. Definitely, hybrid integrated waveguide system may successfully relieve this problem but it is beyond the scope of this paper.

The best approach may be to balance the FCA through adjusting the waveguide structure free from the bandwidth issue. For example, a desired structure may work like this: on one hand, the carrier absorption is increased by carrier injecting, on the other hand, the carrier-induced refractive index reduction can abnormally enhance the optical confinement and decrease the waveguide loss. If the carrier absorption can be compensated by the decreased waveguide loss, the power imbalance resulted from the modulation is removed. The implementation details are still left as an open issue.

## 5. Conclusions

The influence of the loss penalty in a carrier-injected MZI silicon photonic switch was addressed from the basic theory and the practical device fabrication. The CT limits induced by the carrier absorption were given for the conventional design of silicon MZI devices. If a compact and high-crosstalk optical switch is necessary, one should tune the splitter of the first stage dynamically to favor both the two switching states. The paper is intended to provide a roadmap for researchers to design a high-CT, high-speed and wavelength-insensitive silicon optical switch by carrier dispersion effect.

## Acknowledgments

This work is supported by the Natural Basic Research Program of China (No. 2007CB613405), and the Natural Science Foundation of China (No. 60777015).

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