On-chip high-speed optical detection based on an optical rectification scheme in silicon plasmonic platform

Jihua Zhang; Lei Shi; Yilun Wang; Eric Cassan; Xinliang Zhang

doi:10.1364/OE.22.027504

1. Introduction

Optical rectification (OR) is a nonlinear process in which a second-order optical nonlinearity converts an optical field into a direct current (DC) or a radio frequency (RF) voltage. It can be seen as the reverse process of the electro-optic (EO) effect [1, 2]. Reported applications of OR include the generation of terahertz radiations or high-frequency electrical pulses [3–8] and high-speed detection of optical radiations [9–12]. The capability of enabling applications with such a high-speed property is attributed to the instantaneous response of second order nonlinearity. For example, high-frequency electrical pulses up to terahertz scale generated by the OR in poled polymers [4], LiTaO₃ crystals [5] and bulk GaAs [6] have been presented before. Moreover, OR-based photodetectors (ORPD) have many advantages over the traditional semiconductor photodetectors (SPD). Firstly, ORPD can support broadband flat response in optical domain and operate at ultra-high speed (>>100GHz) in electrical domain, while the frequency response of SPD is limited by electron and hole transit times. Actually, the OR efficiency grows as the speed increases. Secondly, no external DC bias is needed in the detection, though the application of a DC bias tends to increase the responsivity, as shown hereafter. Thirdly, as the OR process only converts a fraction of the optical power into an electrical detected signal, the optical output can be reused for other applications. Mikheev et al. experimentally demonstrated an ORPD in nanographite films with a voltage sensitivity of 5 × 10⁻⁷ V/W at hundreds MHz speed [11]. Later, Zhang et al. reported the detection of microwave signals by OR in an integrated AlGaAs waveguide with operation speed to 20 GHz [12]. These schemes were either realized in a free-space configuration or in long distance scales, and most of all were not compatible with silicon photonics which is believed to be the most convenient platform for future integrated optical chips [13].

Based on silicon photonics platform, T. Baehr-Jones et al. experimentally demonstrated an ORPD in a slot-waveguide-based ring resonator with EO polymers filled in the slot, where a current sensitivity of 88 nA/W was observed for a speed of 1 MHz [10]. Unfortunately, this device could not be measured at higher speeds due to substantial output impedance. Recently, the same authors theoretically studied the OR in a silicon slot waveguide instead of considering a ring configuration, in which a much faster response up to 400 GHz was predicted. By using an equivalent circuit model in the analysis, an efficient voltage responsivity of 0.45V/W was envisaged [14]. However, the device length was of 1 mm, i.e. too long for a possible compact on-chip integration application. To overcome this obstacle, plasmonic structures seem to be competitive candidates [15], especially for nonlinear applications [16]. For example, Ward et al. measured the OR in a sub-wavelength plasmonic nanogap but in a free-space configuration [17]. With respect to using integrated silicon plasmonic waveguides, efficient second order nonlinear processes such as EO modulations [18, 19] and frequency conversions [20–23] within short length have been also widely proposed theoretically or demonstrated experimentally due to the large field enhancement and overlap between the interacting frequencies. Another advantage of plasmonic structures lies in the ultra-high speed. Very recently, A. Melikyan et al. demonstrated a high-speed (over 60 GHz) and broadband (beyond 120nm wide near 1550nm) plasmonic phase modulator by filling the plasmonic slot waveguide (PSW) with EO polymers, which is believed to be the most compact high-speed phase modulator demonstrated to date [18].

In this context, a PSW counterpart of the ORPD proposed in Ref [14]. is investigated in the present paper, bringing several advantages: i) the reduction of the device size and thus the ease of group velocity matching between optical frequency (OF) and RF signals due to the short length, ii) the simplification of the electrode design because of the fact that in a PSW the metallic slabs which constitute the waveguide also serve as the electrodes directly, iii) and the improvement of the OR efficiency within short lengths (<23μm) owing to an increased confinement and overlap between OF and RF modes. In addition, instead of using an equivalent circuit model where the assumption of uniform OF and RF mode profiles in the slot area was employed, a more general model directly derived from the Maxwell’s equations is originally presented to calculate the OR voltage. Meanwhile, our model is still able to give similar results as those in Ref [14]. under the same assumption. Specifically, by making use of the optimized PSW structure with a compact footprint of 50nm × 90nm × 22.4μm, an OR voltage responsivity above 1 V/W and an OR current responsivity about 0.42 mA/W are predicted at the RF of 400 GHz. The corresponding normalized OR efficiency is up to 2.25 × 10⁻⁴ W⁻¹. Then the responsivity of the ORPD can be further improved by introducing the electrically induced OR contribution. Moreover, the proposed ORPD supports such an efficient performance over a wide range of light wavelength from 1.25μm to 1.7μm, containing all the classical communication bands. The electrical bandwidth of the ORPD is estimated more than 800 GHz.

2. Device structure

The proposed PSW is shown in Fig. 1. It consists of two metallic slabs on a silicon oxide substrate and separated by a narrow gap filled with an EO polymer. The two metallic slabs also serve as the electrodes in the OR process. Through the OR process, an electrical signal is generated between the two electrodes when an intensity modulated optical signal is injected into the PSW. The length of the PSW is l. The bottom inset shows the cross-section of the PSW, in which the width and height of the slot are defined as w and h, respectively. The EO polymer is considered to be the doped, crosslinked organic polymer with a refractive index of n_p = 1.643 and an EO coefficient of $χ_{111}^{(2)}$ = 619.4 pm/V (r₃₃ = 170 pm/V) at the wavelength of 1550nm [24]. The metal is considered to be silver in forthcoming calculations. In the meantime, results obtained for gold are also given for comparison. It will be shown hereafter that the two metals, which have been chosen due to their better plasmonic properties in the optical domain and higher conductivity in the electrical domain with respect to other ones, present similar voltage responsivities while silver supports larger OR currents and correspondingly larger OR efficiencies. The bottom inset also shows the equivalent circuit at RF for the calculation of the OR current. With respect to the OR mechanism, the PSW is equivalent to a voltage (OR-induced) source V_OR and a C capacitance. Z is the external load resistance. The resistance between the electrodes and the slot has been neglected here due to the high conductivity of the considered metal.

Fig. 1 Schematic of the PSW for investigating the OR effect. The bottom inset shows the cross-section of the PSW and the equivalent electrical circuit at RF. As given in the results section, this PSW with size of w = 50nm, h = 90nm and l = 22.4μm supports an OR voltage responsivity above 1 V/W and an OR current responsivity about 0.42 mA/W at the RF of 400 GHz. The corresponding normalized OR efficiency is up to 2.25 × 10⁻⁴ W⁻¹.

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3. Principle of OR in plasmonic slot waveguides

To begin, let us set the electric field at OF to be $\tilde{E} (t) = E_{ω} e^{- i ω t} + c . c .$ with ω being the angular frequency of the OF. Then, by assuming that the OF signal is an intensity modulated signal with a modulation frequency of ω_RF, the nonlinear polarization resulting from the OR effect for the RF signal is:

P_{R F}^{N L} = 2 ε_{0} χ^{(2)} : E_{ω} E_{ω}^{*} \cos (ω_{R F} t)

Now the RF fields {E_RF, H_RF} fulfill the Maxwell equations with nonlinear polarization:

\begin{array}{l} \nabla \times E_{R F} = - μ \frac{\partial H_{R F}}{\partial t} \\ \nabla \times H_{R F} = ε_{0} ε_{r} \frac{\partial E_{R F}}{\partial t} + \frac{\partial P_{R F}^{N L}}{\partial t} \end{array}

where ε_r is the relative permittivity at RF. We can then obtain:

\nabla \times \nabla \times E_{R F} = μ ε_{0} ε_{r} ω_{R F}^{2} E_{R F} + μ ω_{R F}^{2} P_{R F}^{N L}

When the size of the waveguide is much smaller than the wavelength of the RF signal, i.e. (w, h, l)<<λ_RF, which is satisfied in the proposed configuration, the left part of Eq. (3) is approximately equal to 0. This well-satisfied approximation then leads to:

E_{R F} = - \frac{P_{R F}^{N L}}{ε_{0} ε_{r}} = - \frac{2 χ^{(2)} : E_{ω} E_{ω}^{*}}{ε_{r}} \cos (ω_{R F} t)

This electric field distribution induced from the nonlinear polarization can be expanded over the guided modes, radiation modes, and the evanescent modes at the RF of ω_RF [25]. The most interesting part is the conversion efficiency to the fundamental guided mode due to the largest overlap integral between this mode and the nonlinear RF fields, i.e. E_RF. Defining the fundamental guided mode distribution at ω_RF as {E₀, H₀} corresponding to 1 V voltage between two electrodes, the generated RF voltage value from the nonlinear OR effect is:

V_{0} = \frac{\iint_{g a p} e_{z} \cdot {E_{R F} \times H_{0}} d x d y}{\iint e_{z} \cdot {E_{0} \times H_{0}} d x d y} = - \frac{2 \cos (ω_{R F} t)}{ε_{r}} \frac{\iint_{g a p} e_{z} \cdot {χ^{(2)} : E_{ω} E_{ω}^{*} \times H_{0}} d x d y}{\iint e_{z} \cdot {E_{0} \times H_{0}} d x d y}

For simplicity, we will just consider the x components of the field and nonlinear susceptibility, which is a reasonable assumption in PSW [21]. Further, if we consider the RF field to be uniform in the cross-section of the gap, i.e. $E_{0} = E_{0, x} \cdot e_{x} = 1 [V] / w and H_{0} = H_{0, y} \cdot e_{y}$ , the obtained voltage goes to:

V = - \frac{2 χ_{x x x}^{(2)} \cos (ω_{R F} t)}{ε_{r} h} \iint_{g a p} | E_{ω, x} |^{2} d x d y

Interestingly, if the optical mode is considered to be uniform in the slot as well, then the generated voltage turns into the same value as the Eq. (12) of Ref [14], where a lumped-element equivalent circuit model was employed. In addition, it should be noted that even though the length of the PSW is negligible when compared with the RF wavelength λ_RF, it is usually larger than the optical wavelength λ_ω. As a result, it is needed to take the loss of the optical mode into account. The intensity of the optical mode is attenuated along the propagation in the form of

I_{ω} (z) \propto | E_{ω, x} |^{2} \exp (- α z)

, where α is the attenuation constant of the optical mode. Here, we apply the average value of the intensity along the propagation to calculate the OR-obtained voltage:

V_{O R} = V \frac{\int_{0}^{l} \exp (- α z) d z}{l} = - \frac{2 χ_{x x x}^{(2)} [1 - \exp (- α l)] \cos (ω_{R F} t)}{ε_{r} α h l} \iint_{g a p} | E_{ω, x} |^{2} d x d y

The generated current is then:

I_{O R} = {[\frac{1}{i ω_{R F} C} + Z]}^{- 1} V_{O R} = - \frac{2 χ_{x x x}^{(2)} ω_{R F} ε_{0} [1 - \exp (- α l)] \cos (ω_{R F} t)}{ω_{R F} ε_{0} ε_{r} α h l Z - i α w} \iint_{g a p} | E_{ω, x} |^{2} d x d y

where

C = ε_{0} ε_{r} h l / w

is the capacitance and Z is the external load resistance. As a result, the voltage responsivity V_R, current responsivity I_R and the normalized efficiency of the OR process η in the PSW are:

\begin{matrix} V_{R} = \frac{| V_{O R} |}{P_{ω}} = \frac{2 χ_{x x x}^{(2)} [1 - \exp (- α l)]}{ε_{r} α h l} \frac{\iint_{g a p} | E_{ω, x} |^{2} d x d y}{\iint 2 e_{z} \cdot {E_{ω} \times H_{ω}} d x d y} \\ I_{R} = \frac{| I_{O R} |}{P_{ω}} = \frac{2 χ_{x x x}^{(2)} ω_{R F} ε_{0} [1 - \exp (- α l)]}{\sqrt{{(ω_{R F} ε_{0} ε_{r} α h l Z)}^{2} + α^{2} w^{2}}} \frac{\iint_{g a p} | E_{ω, x} |^{2} d x d y}{\iint 2 e_{z} \cdot {E_{ω} \times H_{ω}} d x d y} \\ η = \frac{| V_{O R} | \cdot | I_{O R} | / 2}{P_{ω}^{2}} = \frac{2 ω_{R F} ε_{0} {χ_{x x x}^{(2)} [1 - \exp (- α l)]}^{2}}{ε_{r} α h l \sqrt{{(ω_{R F} ε_{0} ε_{r} α h l Z)}^{2} + α^{2} w^{2}}} {[\frac{\iint_{g a p} | E_{ω, x} |^{2} d x d y}{\iint 2 e_{z} \cdot {E_{ω} \times H_{ω}} d x d y}]}^{2} \end{matrix}

Here

P_{w} = \iint 2 e_{z} \cdot {E_{ω} \times H_{ω}} d x d y

is the input power of the optical mode. Noting that the parameter of

\iint_{g a p} | E_{ω, x} |^{2} d x d y / \iint 2 e_{z} \cdot {E_{ω} \times H_{ω}} d x d y

is very similar to the interaction factor Γ or the effective index susceptibility γ for the EO effect [26, 27], which makes sense since the OR is considered to be the reversed effect of the EO modulation.

4. Simulation Results

As a starting point, we carry out the mode distributions of the PSW at the OF and RF in the considered structure of Fig. 1. The related field profiles obtained with the finite element based commercial software Comsol Multiphysics are shown in Fig. 2(a) and 2(b), respectively. In Fig. 2(a) the optical power has been normalized to 1W, and in Fig. 2(b) the given mode distribution corresponds to 1V of potential difference between the two electrodes for a RF of 400GHz. The dimensions of w = 50nm and h = 90nm, respectively, have been chosen for the purpose of large OR efficiency after an optimization procedure, as shown soon afterwards. Ascan be seen, both the OF and RF fields are tightly confined in the slot area. Specifically, the electric field at the OF is up to the magnitude of ~10⁸ V/m for 1W, and electric field at the RF arrives to magnitude of ~10⁷ V/m for 1V of drop voltage between the two electrodes. This matter of fact is likely to facilitate the high efficiency of nonlinear processes involving the filling of the slot.

Fig. 2 x-components of the electric field at (a) optical frequency and (b) RF of 400GHz with w = 50nm, h = 90nm. The optical power and the electrical voltage have been normalized to 1W and 1V, respectively.

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Now, based on the calculated modes and Eq. (9), the voltage responsivity V_R, the current responsivity I_R, and the normalized OR efficiency η can be obtained. Figure 3 shows the related results as a function of the slot height for different slot widths. In Fig. 3(a) the ‘5dB length’ is defined as the length where the transmittance of the optical mode is −5dB due to the absorption loss of the metal. The OR results in Fig. 3(b)-(d) are obtained in a PSW with this 5dB length. It is shown that the PSW has larger losses for smaller widths, while the OR shows better performance. This trend can be attributed to the better field confinement at both frequencies for decreasing widths. With respect to the dependence on the waveguide height, the OR voltage decreases and the current increases when the height increases. This phenomenon can be explained as follows. As shown in Eq. (7), the h parameter is in the denominator. Even though the integral term increases for larger heights, the correlation coefficient is smaller than one. In the same time, the impedance of the capacitor goes down as the height goes up, which leads to larger currents. As an interesting result, from Fig. 3(d), we can see that there exists an optimal height for a certain width for the OR efficiency. The obtained optimal heights for different widths are given in Table 1.

Fig. 3 (a) 5dB length (where the transmission of the optical mode is −5dB) of the PSW for different cross-sectional sizes. (b), (c) and (d) are the voltage responsivity, current responsivity and the normalized efficiency of the OR process in the PSW with 5dB length as a function of the height for different values of the width, respectively.

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Table 1. The optimized height and length for different widths and the corresponding voltage responsivity, current responsivity and the optimized normalized efficiency of the OR process^a.

View Table

Next, we set the height to the optimized value and sweep the length of the PSW. The results are shown in Fig. 4. When the length increases, the loss of the optical mode increases, thus reducing the average power along the propagation distance and thus the OR voltage. However, similarly as before, the OR current increases due to the reduction of the capacitor impedance for increasing lengths. As shown in Fig. 4(c), there exists an optimal length as well.

Fig. 4 The voltage responsivity (a), current responsivity (b) and the normalized efficiency (c) of the OR process in the PSW as a function of the PSW waveguide length. Here, the height has been chosen to the optimal value for a certain width coming from the approach giving rise to the results of Fig. 3.

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As a conclusion, there is an optimized structural dimension for the OR efficiency in the PSW to a certain width. In Fig. 5, the black lines mark out the optimized h and l values. In Fig. 5(a) the length has been chosen to be the previously introduced 5dB length, while in Fig. 5(b) the height has been chosen to be the optimized value for each w obtained in Fig. 5(a). Table 1 lists the optimized sizes and results for a couple of specific widths. As a whole, the OR shows better performance for smaller widths. However, it should be noted that the fabrication of the PSW is difficult and the coupling efficiency between silicon waveguides and PSW becomes low when the width is too narrow. Another issue needs to be accounted is that the poling efficiency of the EO polymer may deteriorate in the ultra-narrow slot. Therefore, we choose a smallest width of 50nm, as the efficient coupling between the silicon waveguide and PSW with such a gap was demonstrated in a previous paper in an orthogonal junction configuration [28]. The last row in Table 1 represents the result for the case of gold. Compared with the silver case, the optimized height is similar and so does the generated voltage, while the optimized length is much shorter due to the larger metallic loss. As a result, the OR current and the efficiency are smaller than that by using silver, which is the reason why we mainly choose silver in this analysis. At last, we note that there still exists rooms for further improvement of the OR efficiency in terms of the used EO polymer. For example, in Ref [14]. an EO polymer with r₃₃ = 300 pm/V was considered.

Fig. 5 (a) Normalized OR efficiency as functions of w and h with 5dB length. The black line marks out the optimized h for each w. (b) Normalized OR efficiency as functions of w and l. The height has been chosen to be the optimized value in (a). The black line marks out the optimized l for each w.

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Next, we study the frequency response of this OR mechanism at OF and RF regions. Figure 6 shows the OR property as a function of the optical carrier wavelength for w = 50nm, h = 90nm and l = 22.4μm. As can be seen, the OR in the PSW enables a relatively flat response for a wide wavelength range from 1.25μm to 1.7μm, which contains all the classical communication bands. Figure 6(d) is a plot of the optical mode power transmittance through the 22.4µm long structure, which is around −6dB in the calculated waveband. Figure 7 shows the OR property as a function of the RF. The OR voltage is not given since it is not relevant to the RF according to Eq. (9). We can see that the responsivity of the proposed ORPD is heavily frequency dependent, i.e. it shows a linear dependence on the RF. What should be emphasized here is that a speed limitation still exists in this device. Firstly, the condition of λ_RF >> l must be satisfied for the validity of Eq. (4). If we set this condition to λ_RF ≥ 10l in the PSW, the limiting factor would be f₁ = c/10ln_eff = 815 GHz, where n_eff is the effective refractive index of the RF mode. Here we set n_eff = n_p by considering the metal as an ideal conductor at RF. Beyond this RF range, a travelling-wave model is needed, such as the nonlinear coupled-wave model in [29]. The second factor lies in the transition time of single optical wave-packet in order to avoid the crosstalk between two adjacent wave-packets. This limitation is f₃ = c/ln_g = 5.72 THz, n_g = 2.34 being the group index of the OF mode at λ_ω = 1550nm. Note that there is no RC constant resulted bandwidth limitation in this ORPD. On the contrary, as shown in Fig. 7, the ORPD shows better performance at higher RF. In conclusion, the cut-off frequency is estimated to be around 800 GHz in the present configuration. This high speed is believed to be larger than that in the conventional semiconductor photodetectors by one order of magnitude. This capability of generating ultrahigh speed electrical signals may also derive another potential application of the proposed structure, which is the generation of THz radiation.

Fig. 6 The OR property of the PSW as a function of the optical wavelength for w = 50nm, h = 90nm and l = 22.4μm. (d) is the transmittance of the PSW.

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Fig. 7 The OR property of the PSW as a function of the radio frequency for w = 50nm, h = 90nm and l = 22.4μm.

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In the same time, the current responsivity of proposed ORPD is small when compared with the one in the semiconductor detectors, which have typical responsivities of 0.5 to 1A/W. Apart from the way to employ EO polymer with larger nonlinear coefficients, we propose here another method to increase the responsivity of the ORPD by analyzing the electrical induced OR when introducing a DC electrical bias between the two metallic electrodes. For this case, an effective second order nonlinear coefficient of $χ_{111}^{(2)} + 3 χ_{1111}^{(3)} E_{c}$ is considered when only the x components are taken into account [30]. Here E_c = V_c/w with V_c being the external applied DC voltage. When a typical value of $χ_{1111}^{(3)}$ = 2.3 × 10⁻¹⁹ m²/V² is considered [31], the OR property in the PSW is shown in Fig. 8. It can be seen that the OR process can be linearly controlled and improved by the external applied voltage. The normalized OR efficiency has been further improved more than two times when V_c = 20V. Note that the total voltage is V_c + V_RP_ωcos(ω_RFt) now. Besides, as a common property of nonlinear effect, a high pump power can lead to high OR efficiency. However, it is worth noting that there is a power limitation which comes from two aspects. Firstly, there is an optical damage threshold for the polymer, beyond this value the polymer will get damaged. Secondly, the loss from the metal will elevate the temperature of device. The temperature cannot exceed the glass transition temperature of the used polymer, or the chromophores of the polymer will reorientate and degrade the second-order nonlinear property of the used polymer. Therefore, the input optical power should be carefully chosen in practical experiments.

Fig. 8 The OR property in the PSW as a function of the control DC voltage for w = 50nm, h = 90nm and l = 22.4μm.

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5. Summary and conclusion

Based on an originally proposed and general model, in a PSW with a compact footprint of 50nm × 90nm × 22.4μm, an effective OR process with a normalized efficiency of 2.2517 × 10⁻⁴ W⁻¹ is predicted for an intensity-modulated optical signal with speed of 400GHz. The corresponding voltage and current responsibilities are 1.081V/W and 0.4166mA/W, respectively. Then, an original approach is additionally proposed for increasing the responsivity of the ORPD by bringing in the electrical induced OR. Apart from the classical semiconductor photodetectors that are based on the carrier-induced photoconductive or photovoltaic effects, the proposed ORPD scheme takes advantage of the transient nonlinear optical effect. This makes ORPD supports ultra-high speed (> 800 GHz) around one order of magnitude larger than the one in semiconductor PD. Other advantages include compact footprint and no need for external bias. As to our best knowledge, this work is the first investigation of the nonlinear optical rectification effect in an integrated silicon plasmonic waveguide based on standard silicon-on-insulator technologies. Based on this OR scheme, low-power high-speed broadband optical detection and demodulation can be envisaged in silicon chips.

Acknowledgments

The authors would like to acknowledge the NSFC Major International Joint Research Project (61320106016), the National Science Fund for Distinguished Young Scholars (61125501) and the ANR Program Blanc International POSISLOT (2012, 9035RA13)/NSFC (612111022).

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w (nm)	h (nm)	l (μm)	V_R (V/W)	I_R (mA/W)	η (10⁻⁴/W)
50	90	22.4	1.0810	0.4166	2.2517
80	119	31.5	0.7140	0.3198	1.1417
110	147	39.6	0.5423	0.2743	0.7438
140	174	47.1	0.4421	0.2473	0.5467
50	90	5.4	1.0984	0.1021	0.5607

On-chip high-speed optical detection based on an optical rectification scheme in silicon plasmonic platform

Abstract

1. Introduction

2. Device structure

3. Principle of OR in plasmonic slot waveguides

4. Simulation Results

5. Summary and conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (9)

Optics Express