1. Introduction

Optical modulators are one of the key components for signal transmission and transduction systems, and various types have been reported [1–4]. With the rapidly increasing demand for linear external modulators in recent years, practically viable solutions to improving the modulation linearity are highly desired. Although a number of approaches to improve the linearity of the modulation characteristic have been proposed and applied [5–10], the improvement of the modulation linearity by almost all methods comes at the expense of simplicity of the device design, and up to now no practical device has been found. As we know, the Mach-Zehnder (MZ) interferometer is one of the simplest and most widely used configurations of optical modulators, but the large nonlinear distortion adversely affects its performance and limits its applications, especially in the analog signal processing and transmission systems. The nonlinearity of the MZ modulator is mainly caused by the sinusoidal modulation curve, showing the sublinear characteristic, and so that the linearity may be enhanced by applying super-linear phase modulation to the arm(s) of the MZ interferometer. The microring resonator is a potential and powerful structure that has been used in many applications of optical passive components, such as optical filters and optical dispersion compensators [11–14]. In [15], it was proposed that the microring-based all-pass filter (MRB-APF) could be used to achieve the super-linear characteristic and thus to improve the linearity of the MZ interferometric modulator. Since loss makes heavy impact on the response of the microring resonator, it is very important to clearly know the role of the waveguide loss in the microring-assisted (MRA) MZ intensity modulator. In this paper, the influence of loss on the linearity characteristic of the MRA-MZ modulator is investigated. It will be found that the linearity of the MRA-MZ modulator can still be high at low-loss cases, but phase biasing should be set properly. Large loss will seriously cause power imbalance between two arms of the MZ interferometer and result in limited modulation depth.

2. Basic principle

 

Fig. 1. Schematic diagram of the linearity-enhanced MZ modulator with a microring resonator coupled to one of its arms. The modulation signal is applied to the microring.

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The MRA-MZ intensity modulator is schematically shown in Fig. 1, in which a microring resonator is coupled with one of the arms of the MZ interferometer, forming an all-pass filter (APF). The modulation signal is applied to the microring, modulating the output phase of the all-pass filter and then resulting in the intensity-modulated output of the MZ interferometer. Applying two microrings can form a push-pull configuration, as discussed in [15], but the device operation becomes complicated if the influence of loss is taken into account, so in this paper, we only focus on the single-ring-assisted MZ modulator, as the basic configuration.

The output intensity of the MRA-MZ modulator can be written as:

where I ₀ is the intensity of the input light, σ _0i = exp (-α _0i L _0i) is the amplitude attenuation factor of arm i (=1, 2), and φ _i = β _0i L _0i is the phase delays introduced by arm i (=1, 2). α _0i, β _0i, and L _0i are the amplitude attenuation coefficient, the propagation constant, and the waveguide length of arm i (=1, 2), respectively. The intensity and phase response, $a_r^2$(θ) and φ_r (θ), of the microring-based APF can be written as follows, respectively:

Here $\rho =\sqrt{1-{\kappa }^2}$ is the transmission coefficient and κ is the ring-waveguide amplitude coupling coefficient, σ = exp (-α _r L _r) is the round-trip amplitude attenuation factor, α _r is the amplitude attenuation coefficient of the microring, and θ is the phase delay of the light with a wavelength λ₀ in a round trip L_r of the microring. Generally we can assume that the two arms of the MZ interferometer are identical, i.e. σ ₀₁=σ ₀₂, then Eq. (1) can be rewritten as follows:

where Δφ= φ ₁ - φ ₂ is the phase delay difference of the two arms, which can also be regarded as the phase bias applied on the arms of the MZ interferometer. If waveguide loss is neglectable, i.e. σ ₀₁=σ ₀₂=1 and σ=1, it can be found that the linearity of the MZ modulator can be significantly improved when Δφ is biased to be (m+1/2)π, where m is integer. Owning to the periodicity, we just consider Δφ, θ, and other phase-related parameters in the range from -π to π in the following text.

3. High-order harmonic terms

To assess the linearity of an optical modulator, two criterions are commonly used [16, 5, 9]. One criterion is the level of the spurious signals, also known as the nonlinear distortions, caused by the high-order harmonic terms of the modulation curve. This criterion is commonly used when the optical modulator with a large spur-free dynamic range is expected. In this section, we will analyze the linearity in terms of the high-order harmonic terms of the modulation curve.

For the MRA-MZ modulator, its transfer function given by Eq. (4) can be written in a series of the high-order harmonic terms of Δθ=θ - θ ₀:

where $I^{\left(n\right)}\left(\Delta \theta \right)=\frac{1}{I_0{\sigma }_{01}^2}\frac{d{I}_{\mathit{out}}^n\left(\Delta \theta \right)}{d^n\left(\Delta \theta \right)}$ is the normalized coefficient of the nth-order harmonic term and θ ₀ is the phase bias applied on the microring. If the waveguide loss is ignored, the even-order distortion of the modulation curve given by Eq. (4) can been eliminated by setting Δφ to be π/2 and θ ₀ to be π. Analysis shows that the 3^rd -order term also vanishes when the transmission coefficient ρ is at the value of 0.268 (=2 - √3), and then the lowest high-order distortion will come from the 5^th -order harmonic term.

If the loss of the microring is taken into account, a ² r(θ) is no longer kept unit, and from Eq. (4), it can be found that the modulation curve is no longer antisymmetric around the point π, the biasing point of the loss-free MRA-MZ modulator. The modulation curve with the biasing points Δφ = π/2 and θ₀= π is distorted.

 

Fig. 2. General case of modulation curve (solid) and relocation of the biasing phase θ ₀.

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To analyze the influence of loss on the modulation curve of the MRA-MZ modulator, the biasing points Δφ and θ ₀ should be determined first. We investigated the higher-order terms of Eq. (5) with various transmission coefficients ρ. It is well known that the 2^nd -order distortions can be effectively suppressed by biasing the modulator to the inflection point. For the modulation curve shown in Fig. 2, we choose the center point between θ^Max and θ^Min as the biasing point θ ₀ applied on the microring:

where θ^Max and θ^Min correspond the two points at which the output I _out of the MRA-MZ modulator reaches its maximum and minimum values, respectively. Then, analysis shows that the second-order coefficient I“(0) can always find its zero point at a certain biasing phase Δφ. We choose this value of Δφ as phase bias applied on the arms of the MZ interferometer. Under various transmission coefficients ρ, Δφ that makes I“(0) zero is shown in Fig. 3(a), in which the attenuation factor σ is assumed to be 1.0, 0.8, 0.5, and 0.3, respectively. It can be found that Δφ shift away from π/2, and varies with ρ, while analysis shows that the corresponding biasing point θ ₀ applied on the microring is still at the value of π. Figure 3(b) gives the corresponding 3^rd -order coefficient. From these analytical results, we can conclude that the 2^nd -order terms of the MRA-MZ modulator can always be eliminated by properly setting the biasing phases Δφ and θ ₀.

 

Fig. 3. Phase bias applied on the arms, at which the 2^nd -order harmonic term of the modulation curve vanishes. The corresponding 3^rd-order term is shown in (b).

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From Fig. 3(b), we can find that, for each σ, there always a ρ with which the 3^rd -order coefficient I’’’(0) also vanishes. For comparison, based on Fig. 3(c), we can derive that the 3^rd -order term of the traditional MZ modulator (θ = 0) always exists. We calculated the transmission coefficient θ for various σ, and Fig. 4(a) illustrates the results. The corresponding biasing points Δφ is presented in Fig. 4(b) and θ ₀ is π. The 1^st- and 5^th -order coefficients are illustrated in Fig. 4(c) while the 4^th -order coefficient is at least 7-order smaller than the 1^st -order coefficient and neglectable though the 4^th -order coefficient is no longer zero. With these data, we will be able to design and fabricate the 3^rd-order-free linearized MRA-MZ modulator. We can also calculate the nonlinear distortion caused by the high-order terms of these modulation curves if the MRA-MZ modulator is used in a system.

It should be noted that the 1^st-order coefficient rapidly goes to zero as the loss increases. It means that the transfer efficiency of the modulation signal becomes lower in the lossy cases though the lowest two high-order harmonic terms, the 2^nd - and 3^rd -order terms, can be suppressed. The 4^th-order coefficient I ⁽⁴⁾ is not kept zero any longer at the biasing point if loss exists. It means that the 4^th-order term becomes the lowest high-order distortion source. Certainly, compared to the 5^th -order term, the 4^th -order coefficient is very small.

 

Fig. 4. (a) Transmission coefficient, with which both the 2^nd- and 3^rd-order harmonic terms of the modulation curve vanish, and (b) the corresponding phase bias applied on the arms of the MZ interferometer. The 1^st- and 5^th-order terms are in (c).

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As we know, the fabrication error is unavoidable and it is hard to achieve the MRA-MZ modulator with the right transmission coefficient ρ and attenuation factor σ presented in Fig. 4(a). However, based our analysis, we found that there is always a pair of biasing phases Δφ and θ ₀ at which the 2^nd - and 3^rd -order terms of the MRA-MZ modulator can be zero. Certainly, this biasing point θ ₀ is not at the center point between θ^Max and θ^Min , and so the modulation efficiency will be limited.

4. Range of linearity

The linearity range of the modulation curve [5, 9] is another quantitative criterion. The linearity range, in fact, is the maximum modulation depth at which the deviation of the modulation curve from the best linear fit is still lower than a specified value η. It is a very important parameter to evaluate the lightwave transmission efficiency of an analog system. Assuming that the best linear fit is f = f(Δθ), we have the following expression of the linearity range m:

where $I_{\mathit{\text{out}}}^U$ and $I_{\mathit{\text{out}}}^L$ are the output intensities corresponding to the upper and lower limits Δθ ^U and Δθ ^L of the phase change range within which the following deviation condition is satisfied:

Here, as shown in Fig. 2, Δ$I_{\mathit{\text{out}}}^{\mathit{\text{Max}}}$ = $I_{\mathit{\text{out}}}^{\mathit{\text{Max}}}$ - $I_{\mathit{\text{out}}}^{\mathit{\text{Min}}}$ , and $I_{\mathit{\text{out}}}^{\mathit{\text{Max}}}$ and $I_{\mathit{\text{out}}}^{\mathit{\text{Min}}}$ are the maximum and minimum intensities of the output, respectively. Commonly η was specified as 1% [5, 9].

Setting Δφ = π/2 and θ ₀= π, we calculated the linearity range of the MRA-MZ modulator at the no loss case. Figure 5(a) shows the calculation result. It can be found that the linearity range is always larger than 90% when ρ is set between 0.25 to 0.42 and the maximum value can even reach up to 99.5% at ρ ≈ 0.42. In comparison, when ρ is zero, the linearity range is only about 72%, which is the value of the traditional MZ modulator. Figure 5(b) illustrates the modulation curve of the MRA-MZ modulator with ρ ≈ 0.35 and σ = 1.0. The best linear fit f(Δθ) = 0.5-0.485Δθ and the deviation [I_out (Δθ)- f(Δθ)]/$I_{\mathit{\text{out}}}^{\mathit{\text{Max}}}$ of the modulation curve are also presented in this figure. The linearity range of this modulation curve is 97.3%.

As analyzed in the last section, if the waveguide loss is taken into account, the modulation curve around the biasing points Δφ = π/2 and θ ₀= π is distorted. To acquire higher modulation linearity, the biasing points Δφ and θ ₀ should be relocated. The method to eliminate the 2^nd -order harmonic term of the modulation curve, as introduced in the last section, is used again to obtain the values of Δφ and θ ₀ for the MRA-MZ modulator with various loss and microring coupling coefficients. Based on the data given by Fig. 3, the linearity range of the MRA-MZ modulator with various transmission coefficients ρ is calculated and illustrated in Fig. 6. It can be found that the maximum linearity range decreases as the loss increases. However, the linearity range is kept above 90% within a rather large range of the transmission coefficient ρ if the attenuation factor is larger than 0.7, which corresponds to the round-trip loss of about 3 dB. This data shows that the fabrication tolerance of high-linearity MRA-MZ modulators is large if the loss can be controlled at a low level.

 

Fig. 5. (a) Linearity range m of the MRA-MZ modulator at the loss-free case. (b) Modulation curve of the MRA-MZ modulator with the transmission coefficient of 0.35 and the deviation.

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Fig. 6. Linearity range versus transmission coefficient under different attenuation factors.

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Fig. 7. (a) Maximum linearity range m and m’ of the MRA-MZ modulator with various attenuation factors, and (b) the corresponding transmission coefficient. The corresponding phase bias applied on the arms of the MZ interferometer is in (c).

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The solid line in Fig. 7(a) presents the maximum linearity range m of the MRA-MZ modulator with various microring losses. The corresponding ρ and Δφ are presented in Figs. 7(b) and (c), respectively. It can be found that m decreases as the loss increases. Since the amplitude response of the MRB-APF will attenuated severely if the microring loss exists, causing the power imbalance between the two arms of the MZ interferometer, the maximum modulation depth, defined as ${\mathit{MD}}^{\mathit{Max}}=\frac{\Delta I_{\mathit{out}}^{\mathit{Max}}}{I_{\mathit{out}}^{\mathit{Max}}},$, of the transfer function given by Eq. (4) will becomes less than unit. It means that loss will introduce an intrinsic limitation to the linearity range of the MRA-MZ modulator. Replacing the denominator of Eq. (7) with ∆$I_{\mathit{\text{out}}}^{\mathit{\text{Max}}}$ and defining $m\text{'}=\frac{I_{\mathit{out}}^U-I_{\mathit{out}}^L}{{\Delta I}_{\mathit{out}}^{\mathit{Max}}}$, we calculated m’, as shown by the dash line in Fig. 7(a) and found that m’ is always larger than 99% when σ is larger than 0.4. It means the intrinsic maximum modulation depth of the modulation curve is the main factor that limits the linearity range m.

 

Fig. 8. Modulation curve of the MRA-MZ modulator with (a) the imbalance ratio ζ = 1.4 and (b) the balance ratio ζ =1.0.

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To improve the linearity range m, we can intentionally set the power imbalanced between the two arms of the MZ interferometer, which can be realized by properly setting the splitting ratio of the two Y-branch couplers, and then Eq. (4) should be revised as

where ζ is the power imbalance factor introduced by the unequally splitting ratio of the two Y-branch couplers of the MZ interferometer. From Eq. (9), it can be noted that the intrinsic maximum modulation depth can be enlarged back to unit by increasing the factor ζ. Figure 8(a) shows an example, in which ζ is 1.4. Though the attenuation factor is 0.5, corresponding to the round-trip loss of about 6dB, the linearity range can still reach up to 98.6%. For comparison, the modulation curve of the MRA-MZ modulator with the balanced power ratio is illustrated in Fig. 8(b), in which the linearity range is only 77.2% while its m’ is 94.9%.

5. Conclusion

In this paper, the influence of the waveguide loss on the linearity of the MRA-MZ modulator is investigated. The linearity of the MRA-MZ modulator is greatly high, even at the low-loss case. The 2^nd- and 3^rd-order terms of the MRA-MZ modulator can be eliminated by properly setting the biasing phases Δφ and θ ₀, and the maximum linearity range is larger than 90% when the attenuation factor σ is larger than 0.7. Analysis shows that the maximum modulation depth is the main factor limiting the linearity range of the modulation curve. By properly setting the power ratio between the two arms of the MZ interferometer, the intrinsic maximum modulation depth can be improved and the linearity range can be kept large even at large loss conditions.