Tunable Band-Pass Filters with Long Periodicity Using Cascaded Mach-Zehnder Interferometer Networks

Abstract

This paper introduces a theoretical framework for designing and tuning band-pass filters with a highly extended periodicity using cascaded Mach-Zehnder Interferometer (MZI) networks. We show that a filter centered at frequency f₀ with a bandwidth of FSR₀ and an arbitrarily large free spectral range (FSR) can be built with a minimal number of MZIs by using stages with FSRs that are prime multiples of FSR₀. Due to the inherent multi-spectral transparency of materials, this design ensures that only a single narrow passband is transparent. We derive the total power transmission for such a cascaded system and show that the filter’s overall periodicity is the product of the individual MZI transfer functions. Furthermore, we deduce the linear relationship between the applied differential voltage and the resulting frequency shift, offering a precise method for continuous spectral tuning without altering the filter’s intrinsic FSR. We propose a new, simplified electronic circuit that uses a single input current and series impedances for continuous resonant peak tuning and analyze the feasibility of such a design. This circuit improves practical implementation and allows for compensation of fabrication errors. This work offers crucial analytical tools and insights for developing advanced reconfigurable photonic integrated filters, essential for future optical communication and sensing systems.

1. Introduction

MZIs are fundamental components in integrated photonics, serving as critical elements in devices such as optical switches, high-speed modulators, and wavelength filters. Their repeating spectral behavior makes them particularly well-suited for wavelength division multiplexing (WDM) systems, notably as optical interleavers [1]. Recent advances in hybrid photonic integrated circuit (PIC) technologies, merging platforms such as Indium Phosphide (InP) and silicon nitride (Si₃N₄) TriPleX^TM waveguides [2,3], have significantly improved their capabilities. TriPleX^TM technology, for example, offers exceptionally low propagation losses (<$0.1$ dB/cm) and tight bending radii (<60 µm), facilitating high levels of planar integration for complex optical circuits.

The ongoing demand for enhanced spectral efficiency and flexibility in optical networks has spurred the development of more sophisticated filter designs. MZI cascades (Figure 1), where each stage incorporates a distinct FSR or other unique spectral properties, enable the creation of highly specific filter responses. These include wider and flatter passbands, superior rejection of unwanted signals, and precise channel selection [4]. In the context of microwave photonics, MZIs are crucial for functionalities like single-sideband filtering and integrated beamforming networks, as seen in applications like the SPACEBEAM project’s Photonic Integrated Beamforming Receiver for Scan-on-Receive Synthetic Aperture Radar (SCORE-SAR) (described in [3]). Here, InP Mach-Zehnder Modulators (MZMs) with low half-wave voltages (as low as 3 V) and high modulation frequencies (up to 40 GHz) are integrated with TriPleX™ passive structures.

These sophisticated filter responses are often achieved by cascading MZIs with different FSRs to harness the Vernier effect [5,6]. The Vernier effect arises when two periodic functions are combined; when MZI filters with different FSR periodicities are concatenated, the resulting transmission maxima are separated. This approach can enhance the signal-to-noise ratio and is widely used for high-sensitivity sensing by comparing signals of similar wavelengths. The filter design presented in this work can be conceptualized as an arbitrary cascade of filters, each possessing a co-aligned transmission peak at a specific central frequency. By leveraging the Vernier effect, the distinct periodicities of each filter stage ensure that no other constructive interference maxima occur within the operational transparency region of the device.

Our design is based on a cascaded MZI configuration, similar to other works that aim for tunability [7], but is specifically engineered to tune a single passband. The key achievement is an aperiodic tunable transparency. This architecture is modulator-agnostic, requiring only that each stage can apply a phase shift via a controllable physical parameter. As an example of its application, this paper will particularize the design for electro-optic (EO) modulators, although the specific EO coefficient or material (such as doped silicon or Lithium Niobate (LN)) is not fundamental to the underlying theory.

Other prominent integrated filter structures include phase-shifted Bragg gratings (PS-BGs) and ring resonators (RRs). RRs, in particular, are known for their compact footprint and are often used in cascaded Vernier configurations for sensing or as widely tunable semiconductor lasers [5,8]. Recent works on thin-film LN platforms have demonstrated tunable notch filters and ring-pair structures that are highly compact [9,10]. While many of these advanced designs couple RRs with MZIs to achieve their filtering goals [7,11], which can offer a more compact footprint, our proposed design relies purely on cascaded MZIs to obtain aperiodic tuning.

Arbitrarily large-periodicity MZI cascades introduce greater design freedom but also necessitate precise control and a thorough understanding of the network’s overall transfer function. Such complex designs often require a significant number of phase controllers, for instance, up to 72 for a 3 × 12 optical Blass matrix.

It is important to justify this configuration in contrast to more common MZI applications. Typically, MZI filters are cascaded to produce periodic frequency filters, often called optical frequency combs. As previously mentioned, if the filters have different periodicities, the Vernier effect is produced [5,6]. The configuration presented here, however, allows for the creation of virtually aperiodic filters with a single window of transparency. This window can be tuned, and it is possible to do so while preserving the filter’s spectral shape. While any optical modulator can be used, this work also presents a mechanism to synchronize the modulation of each stage using a single control variable.

Beyond fixed designs, dynamic tunability is paramount for reconfigurable optical networks, adaptable sensors [12], and microwave photonic filters [13,14]. The ability to shift the central frequency of a complex filter network using a single control, while preserving its spectral shape, is a key capability [15]. This tunability is often achieved through thermo-optic phase shifters, leveraging mechanisms like metal heaters deposited on waveguides, or by dynamically adjusting the MZI’s FSR in sensing applications, such as temperature measurement [16]. Furthermore, advanced MZI-based architectures, sometimes combined with ring resonators [8,11], are being explored for dispersion compensation and to improve the dynamic range of microwave photonic filters through techniques like carrier suppression.

While many filter types exist across multiple technologies, the primary motivation of this study is to deduce and formally state the necessary conditions to achieve a single, virtually aperiodic passband filter using cascaded MZIs. We derive the conditions for continuous, shape-preserving tuning and propose a novel control design for its implementation, with a specific application example using electro-optic modulators. This paper is organized as follows: Section 2 derives the complex amplitude transfer function for a single asymmetric MZI, including losses and non-ideal couplers. Section 3 describes how to design arbitrarily large-periodicity band-pass filters, covering the minimum number of MZIs needed, and the total power transmission. Section 4 formally shows the linear relationship between frequency and the voltage applied for tuning with dual modulators. Section 5 introduces a practical circuit for continuous resonant peak tuning and Section 6 gives insights for the feasibility of that design.

2. Theoretical Framework of Mach-Zehnder Interferometers

Throughout this paper, we consider the propagation of optical fields. The phase accumulation in a waveguide is governed by the effective refractive index ($n_{\mathrm{eff}}$), while the Free Spectral Range (FSR) and frequency shifts due to path length differences are determined by the group refractive index ($n_{\mathrm{g}}$) as they relate to group delay [17]. For simplicity in our derivations and where not explicitly differentiated, we assume the waveguides exhibit low dispersion, implying $n_{\mathrm{eff}}\approx n_{\mathrm{g}}$.

2.1. Single Asymmetric MZI with Losses and Non-Ideal Splitters

An MZI is made of two optical couplers joined by two waveguide arms [4]. We examine an MZI that is asymmetric and includes various non-ideal characteristics, as described in the system model below.

2.1.1. System Model

Consider an MZI with the following parameters:

• Input optical field: $E_{\mathrm{in}}$.
• Input waveguide amplitude loss coefficient: ${\alpha }_{\mathrm{in}}$.
• Output waveguide amplitude loss coefficient: ${\alpha }_{\mathrm{out}}$.
• Arm 1: physical length $L_1$, amplitude loss coefficient ${\alpha }_1$, propagation constant ${\beta }_1$.
• Arm 2: physical length $L_2$, amplitude loss coefficient ${\alpha }_2$, propagation constant ${\beta }_2$.
• First coupler (input splitter) power coupling ratio: $\eta$ to arm 1, $(1-\eta )$ to arm 2.
• Second coupler (output combiner): Assumed to be a lossless 3 dB (50:50) splitter.

2.1.2. Input Attenuation

The optical field entering the first coupler, $E_0$, is attenuated by the input waveguide loss:

$$\begin{array}{c}\hfill E_0=E_{\mathrm{in}}{e}^{-{\alpha }_{\mathrm{in}}/2}\end{array}$$

(1)

2.1.3. First Coupler (Non-Ideal Splitter)

Assuming a lossless input coupler with power splitting ratio $\eta$ to arm 1 and $1-\eta$ to arm 2, the complex amplitudes after the first coupler are:

$$\begin{array}{cc}\hfill E_1& =\sqrt{\eta }{E}_0\hfill \\ \hfill E_2& =j\sqrt{1-\eta }{E}_0\hfill \end{array}$$

(2)

(A phase factor j is typically included for the cross-port output to ensure a unitary transformation for a directional coupler, though this can be generalized based on coupler design.)

2.1.4. Propagation Through Arms

As the fields propagate through the MZI arms, they experience both amplitude loss and phase shift. The complex amplitudes at the input of the second coupler are:

$$\begin{array}{cc}\hfill E_1^{\prime }& =\sqrt{\eta }{E}_0{e}^{-{\alpha }_1{L}_1/2}{e}^{-j{\beta }_1{L}_1}\hfill \\ \hfill E_2^{\prime }& =j\sqrt{1-\eta }{E}_0{e}^{-{\alpha }_2{L}_2/2}{e}^{-j{\beta }_2{L}_2}\hfill \end{array}$$

(3)

2.1.5. Second Coupler (Assumed 50:50 for Simplicity)

The second coupler combines the fields from the two arms. For a lossless 3 dB coupler, the transfer matrix is:

$$\left[\begin{array}{c}{E}_{\mathrm{out},1}\\ E_{\mathrm{out},2}\end{array}\right]={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& j\\ j& 1\end{array}\right]\left[\begin{array}{c}{E}_1^{\prime }\\ E_2^{\prime }\end{array}\right]$$

(4)

The complex amplitude at Output Port 1 ($E_{\mathrm{out},1}$) is:

$$\begin{array}{cc}\hfill E_{\mathrm{out},1}& ={\displaystyle \frac{1}{\sqrt{2}}}(E_1^{\prime }+j{E}_2^{\prime })\hfill \\ \hfill & ={\displaystyle \frac{E_0}{\sqrt{2}}}\left(\sqrt{\eta }{e}^{-{\alpha }_1{L}_1/2}{e}^{-j{\beta }_1{L}_1}+\right.\hfill \\ \hfill & \left.j\left(j\sqrt{1-\eta }{e}^{-{\alpha }_2{L}_2/2}{e}^{-j{\beta }_2{L}_2}\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{E_0}{\sqrt{2}}}\left(\sqrt{\eta }{e}^{-{\alpha }_1{L}_1/2}{e}^{-j{\beta }_1{L}_1}-\right.\hfill \\ \hfill & \left.\sqrt{1-\eta }{e}^{-{\alpha }_2{L}_2/2}{e}^{-j{\beta }_2{L}_2}\right)\hfill \end{array}$$

(5)

2.1.6. Output Waveguide Loss

The final output field $E_{\mathrm{out}}$ after traversing the output waveguide is:

$$\begin{array}{cc}\hfill E_{\mathrm{out}}& =E_{\mathrm{out},1}{e}^{-{\alpha }_{\mathrm{out}}/2}\hfill \end{array}$$

(6)

2.1.7. Total Transfer Function

Let us define the complex amplitudes for each arm at the second coupler input, normalized to $E_0$:

$$\begin{array}{cc}\hfill A_1& =\sqrt{\eta }{e}^{-{\alpha }_1{L}_1/2}{e}^{-j{\beta }_1{L}_1}\hfill \\ \hfill A_2& =j\sqrt{1-\eta }{e}^{-{\alpha }_2{L}_2/2}{e}^{-j{\beta }_2{L}_2}\hfill \end{array}$$

(7)

And define the amplitude attenuation terms for propagation:

$$\begin{array}{cc}\hfill {\mathcal{A}}_1& =\sqrt{\eta }{e}^{-{\alpha }_1{L}_1/2}\hfill \\ \hfill {\mathcal{A}}_2& =\sqrt{1-\eta }{e}^{-{\alpha }_2{L}_2/2}\hfill \end{array}$$

(8)

The total phase difference between the arms is $\Delta \varphi ={\beta }_2{L}_2-{\beta }_1{L}_1$. The total complex amplitude transfer function $T\left(\omega \right)=E_{\mathrm{out}}/E_{\mathrm{in}}$ for Output Port 1 is:

$$\begin{array}{cc}\hfill T\left(\omega \right)& ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})/2}}{\sqrt{2}}}\left({\mathcal{A}}_1{e}^{-j{\beta }_1{L}_1}-{\mathcal{A}}_2{e}^{-j{\beta }_2{L}_2}\right)\hfill \end{array}$$

(9)

2.1.8. Power Transmission

The power transmission ${\left|T\left(\omega \right)\right|}^2$ at Output Port 1 is:

$$\begin{array}{cc}\hfill {\left|T\left(\omega \right)\right|}^2& =\hfill \\ \hfill & {\left|{\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})/2}}{\sqrt{2}}}\left({\mathcal{A}}_1{e}^{-j{\beta }_1{L}_1}-{\mathcal{A}}_2{e}^{-j{\beta }_2{L}_2}\right)\right|}^2\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}{\left|{\mathcal{A}}_1{e}^{-j{\beta }_1{L}_1}-{\mathcal{A}}_2{e}^{-j{\beta }_2{L}_2}\right|}^2\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}\left[{\left({\mathcal{A}}_1\right)}^2+{\left({\mathcal{A}}_2\right)}^2-\right.\hfill \\ \hfill & \left.2{\mathcal{A}}_1{\mathcal{A}}_{2}cos({\beta }_1{L}_1-{\beta }_2{L}_2)\right]\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}\left[\eta e^{-{\alpha }_1{L}_1}+(1-\eta )e^{-{\alpha }_2{L}_2}-\right.\hfill \\ \hfill & \left.2\sqrt{\eta (1-\eta )}{e}^{-({\alpha }_1{L}_1+{\alpha }_2{L}_2)/2}cos(\Delta \varphi )\right]\hfill \end{array}$$

(10)

The power transmission $|T_2{\left(\omega \right)|}^2$ at Output Port 2 is derived similarly from $E_{\mathrm{out},2}={\displaystyle \frac{1}{\sqrt{2}}}(j{E}_1^{\prime }+E_2^{\prime })$:

$$\begin{array}{cc}\hfill & |T_2{\left(\omega \right)|}^2=\hfill \\ \hfill & {\left|{\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})/2}}{\sqrt{2}}}\left(j{\mathcal{A}}_1{e}^{-j{\beta }_1{L}_1}+{\mathcal{A}}_2{e}^{-j{\beta }_2{L}_2}\right)\right|}^2\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}{\left|j{\mathcal{A}}_1{e}^{-j{\beta }_1{L}_1}+{\mathcal{A}}_2{e}^{-j{\beta }_2{L}_2}\right|}^2\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}\left[{\left({\mathcal{A}}_1\right)}^2+{\left({\mathcal{A}}_2\right)}^2+\right.\hfill \\ \hfill & \left.2{\mathcal{A}}_1{\mathcal{A}}_{2}sin({\beta }_1{L}_1-{\beta }_2{L}_2)\right]\hfill \\ \hfill & ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}+{\alpha }_{\mathrm{out}})}}{2}}\left[\eta e^{-{\alpha }_1{L}_1}+(1-\eta )e^{-{\alpha }_2{L}_2}+\right.\hfill \\ \hfill & \left.2\sqrt{\eta (1-\eta )}{e}^{-({\alpha }_1{L}_1+{\alpha }_2{L}_2)/2}sin(\Delta \varphi )\right]\hfill \end{array}$$

(11)

where $\Delta \varphi ={\beta }_2{L}_2-{\beta }_1{L}_1$.

2.2. Cascaded MZI Networks

For an interleaver network, multiple MZI stages are cascaded. Each stage $i\in \{1,2,\dots ,N\}$ may have its own set of parameters as defined previously, denoted with a superscript $\left(i\right)$. Furthermore, each arm within an MZI stage is equipped with an optical phase modulator.

2.2.1. Per-Stage Transfer Function

For each MZI stage i, with a general coupling ratio ${\eta }^{\left(i\right)}$, the complex field transfer function for Output Port 1 is:

$$\begin{array}{cc}\hfill T^{\left(i\right)}\left(f_m\right)& ={\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}^{\left(i\right)}+{\alpha }_{\mathrm{out}}^{\left(i\right)})/2}}{\sqrt{2}}}\hfill \\ \hfill & \left({\mathcal{A}}_1^{\left(i\right)}{e}^{-j{\tilde{\beta }}_1^{\left(i\right)}(f_m,V_1^{\left(i\right)})}-{\mathcal{A}}_2^{\left(i\right)}{e}^{-j{\tilde{\beta }}_2^{\left(i\right)}(f_m,V_2^{\left(i\right)})}\right)\hfill \end{array}$$

(12)

where ${\mathcal{A}}_1^{\left(i\right)}$ and ${\mathcal{A}}_2^{\left(i\right)}$ are the amplitude attenuation terms for stage i, and ${\tilde{\beta }}_1^{\left(i\right)}$ and ${\tilde{\beta }}_2^{\left(i\right)}$ incorporate the phase shifts due to applied voltages.

2.2.2. Total Transfer Function for Cascaded Networks

Assuming a serial connection of N stages, where the output of one stage is the input to the next, and neglecting additional interconnect losses between stages, the total complex field transfer function for the entire network is the product of the individual stage transfer functions [18]:

$$\begin{array}{c}\hfill T_{\mathrm{total}}\left(f_m\right)=\prod _{i=1}^N{T}^{\left(i\right)}\left(f_m\right)\end{array}$$

(13)

2.2.3. Total Power Transmission

The total power transmission of the cascaded network at a given frequency $f_m$ is the product of the individual stage power transmissions:

$$\begin{array}{c}\hfill |T_{\mathrm{total}}\left(f_m\right){|}^2=\prod _{i=1}^N{\left|T^{\left(i\right)}\left(f_m\right)\right|}^2\end{array}$$

(14)

This describes the complete complex transmission response of N cascaded asymmetric MZIs.

3. Arbitrarily Large-Periodicity Band-Pass Filter Design and Characteristics

This section addresses the design principles for generating a band-pass filter with a specified width (FSR₀) and an arbitrarily large periodicity using a minimum number of cascaded MZIs. We formally deduce the total power transmission and prove the overall periodicity.

3.1. Minimum MZI Count for Arbitrary Periodicity

A single MZI functions as a periodic filter with an FSR inversely proportional to the optical path length difference ($\Delta L$). Specifically, $\mathrm{FSR}=c/(n_g\Delta L)$, where c is the speed of light in vacuum and $n_g$ is the effective refractive index.

To achieve an arbitrary large periodicity for a band-pass filter of width FSR₀, we employ a cascade of N MZIs, where the FSRs of the individual MZIs are chosen as prime multiples of FSR₀. Let FSR_i denote the FSR of the i-th MZI. We select ${\mathrm{FSR}}_i=p_i{\mathrm{FSR}}_0$, where $p_i$ are distinct prime numbers.

The composite FSR of a cascaded system of filters is given by the least common multiple (LCM) of the individual FSRs. If the individual FSRs are ${\mathrm{FSR}}_1,{\mathrm{FSR}}_2,\dots ,{\mathrm{FSR}}_N$, then the total FSR of the cascaded system is ${\mathrm{FSR}}_{\mathrm{total}}=\mathrm{LCM}({\mathrm{FSR}}_1,{\mathrm{FSR}}_2,\dots ,{\mathrm{FSR}}_N)$.

For our design, ${\mathrm{FSR}}_i=p_i{\mathrm{FSR}}_0$. Thus,

$$\begin{array}{c}\hfill {\mathrm{FSR}}_{\mathrm{total}}=\mathrm{LCM}(p_1{\mathrm{FSR}}_0,p_2{\mathrm{FSR}}_0,\dots ,p_N{\mathrm{FSR}}_0)\end{array}$$

(15)

Since $p_1,p_2,\dots ,p_N$ are distinct prime numbers, their LCM is simply their product:

$$\begin{array}{c}\hfill \mathrm{LCM}(p_1,p_2,\dots ,p_N)=p_1{p}_2\dots p_N\end{array}$$

(16)

Therefore, the total FSR of the cascaded system becomes:

$$\begin{array}{c}\hfill {\mathrm{FSR}}_{\mathrm{total}}=(p_1{p}_2\dots p_N){\mathrm{FSR}}_0\end{array}$$

(17)

Let FSR₀ define the target passband width of the composite filter. We design one MZI stage (e.g., stage 1) to have its FSR equal to FSR₀, while subsequent stages have FSRs chosen as prime multiples of FSR₀. Then, the overall passband will align to ${\mathrm{FSR}}_0$. By choosing a sufficiently large number of distinct prime numbers N, the product $P=p_1{p}_2\dots p_N$ can be made arbitrarily large. This allows the overall filter periodicity to be arbitrarily large, effectively creating an arbitrarily large-periodicity (“quasi-aperiodic”) filter over a wide spectral range (Figure 2).

The minimum number of MZIs required is $N\ge 1$. To achieve an arbitrarily large periodicity, N must be chosen such that the product of N distinct prime numbers is sufficiently large to meet the desired arbitrarily large periodicity, while ensuring at least one MZI has a primary FSR of FSR₀ to define the filter’s fundamental bandwidth. For practical implementation, each MZI has a $\Delta L_i$ such that $\Delta L_i=\Delta L_0/p_i$ for an FSR of $p_i{\mathrm{FSR}}_0$, where ${\mathrm{FSR}}_0=c/(n_g\Delta L_0)$.

3.2. Total Power Transmission of the Designed Filter

Equation (13) and Equation (14) are the total complex field transfer function $T_{\mathrm{total}}\left(f_m\right)$ and the total power transmission $|T_{\mathrm{total}}|$ respectively. For each MZI stage i, its power transmission is given by:

$$\begin{array}{cc}\hfill |T^{\left(i\right)}& \left(f_m\right){|}^2=\hfill \\ \hfill & {\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}^{\left(i\right)}+{\alpha }_{\mathrm{out}}^{\left(i\right)})}}{2}}\hfill \\ \hfill & \left[{\eta }^{\left(i\right)}{e}^{-{\alpha }_1^{\left(i\right)}{L}_1^{\left(i\right)}}\right.\hfill \\ \hfill & +(1-{\eta }^{\left(i\right)})e^{-{\alpha }_2^{\left(i\right)}{L}_2^{\left(i\right)}}\hfill \\ \hfill & -2\sqrt{{\eta }^{\left(i\right)}(1-{\eta }^{\left(i\right)})}{e}^{-({\alpha }_1^{\left(i\right)}{L}_1^{\left(i\right)}+{\alpha }_2^{\left(i\right)}{L}_2^{\left(i\right)})/2}\hfill \\ \hfill & \left.cos\left(\Delta {\varphi }^{\left(i\right)}{f}_m\right)\right]\hfill \end{array}$$

(18)

Therefore, the total power transmission for the entire cascaded filter network is (see Figure 3 for the theoretical result and Figure 4 for the simulation with Lumerical INTERCONNECT):

$$\begin{array}{cc}\hfill |T_{\mathrm{total}}& \left(f_m\right){|}^2=\hfill \\ \hfill & \prod _{i=1}^N\left\{{\displaystyle \frac{e^{-({\alpha }_{\mathrm{in}}^{\left(i\right)}+{\alpha }_{\mathrm{out}}^{\left(i\right)})}}{2}}\left[{\eta }^{\left(i\right)}{e}^{-{\alpha }_1^{\left(i\right)}{L}_1^{\left(i\right)}}+\right.\right.\hfill \\ \hfill & (1-{\eta }^{\left(i\right)})e^{-{\alpha }_2^{\left(i\right)}{L}_2^{\left(i\right)}}-\hfill \\ \hfill & 2\sqrt{{\eta }^{\left(i\right)}(1-{\eta }^{\left(i\right)})}{e}^{-({\alpha }_1^{\left(i\right)}{L}_1^{\left(i\right)}+{\alpha }_2^{\left(i\right)}{L}_2^{\left(i\right)})/2}\hfill \\ \hfill & \left.\left.cos\left({\displaystyle \frac{2\pi n_{\mathrm{eff}}\Delta L^{\left(i\right)}{f}_m}{c}}+{\varphi }_{i,\mathrm{initial}}+\Delta {\varphi }_{i,\mathrm{tune}}\right)\right]\right\}\hfill \end{array}$$

(19)

where i are the primes between 1 and the N-th prime number.

This equation defines the total power transmission of the band-pass filter as a function of frequency, considering individual MZI parameters and differential tuning phase shifts.

The novelty of this architecture lies in its ability to achieve a narrow passband, its compatibility with high-speed linear RF modulators, and its capacity for an arbitrarily high passband-to-rejection ratio.

This design methodology minimizes control complexity. By establishing specific scaling relationships, the subsequent tuning of the passband is simplified, requiring only a minimum number of MZIs. Furthermore, the inclusion of modulators provides a robust mechanism for compensating for fabrication tolerances and thermal effects, as the filter can be finely tuned and adjusted via an external control parameter.

Thus far, we have established the conditions for the filtering effect, which depends solely on the MZI combination. When these MZIs are implemented using modulators, it becomes possible to alter the filter’s response. We will now determine the specific conditions required to continuously tune the filter, while preserving its spectral shape, by incorporating electro-optic modulators into the MZI arms (Figure 5).

4. Linear Relationship of Frequency with Applied Voltage

This section deduces the linear relationship between the frequency of the resonant peak of an MZI filter and the applied differential tuning voltage. This relationship is important for tuning of the filter’s passband while maintaining its intrinsic FSR (Figure 6).

4.1. Differential Phase Shift Model with Dual Modulators

For an MZI where both arms are equipped with electro-optic phase modulators, the induced phase shifts depend on the applied voltages. Let $V_1^{\left(i\right)}$ be the voltage applied to the modulator in arm 1 of MZI stage i, and $V_2^{\left(i\right)}$ be the voltage applied to the modulator in arm 2. Assuming a linear electro-optic effect, the phase shifts are:

$$\begin{array}{c}\hfill {\varphi }_{app,1}^{\left(i\right)}\left(V_1^{\left(i\right)}\right)={\alpha }^{\prime }{V}_1^{\left(i\right)}{L}_{mod,1}^{\left(i\right)}\end{array}$$

(20)

$$\begin{array}{c}\hfill {\varphi }_{app,2}^{\left(i\right)}\left(V_2^{\left(i\right)}\right)={\alpha }^{\prime }{V}_2^{\left(i\right)}{L}_{mod,2}^{\left(i\right)}\end{array}$$

(21)

where ${\alpha }^{\prime }$ is the electro-optic efficiency constant, and $L_{mod,1}^{\left(i\right)}$ and $L_{mod,2}^{\left(i\right)}$ are the effective lengths of the modulators in arms 1 and 2, respectively. For simplicity, we assume $L_{mod,1}^{\left(i\right)}=L_{mod,2}^{\left(i\right)}=L_{mod}^{\left(i\right)}$ across both arms of a given MZI stage.

The total phase difference ${\Phi }^{\left(i\right)}(f,V_1^{\left(i\right)},V_2^{\left(i\right)})$ between the arms of MZI stage i is the sum of the intrinsic phase difference due to path length imbalance and the differential applied phase shift:

$$\begin{array}{cc}\hfill {\Phi }^{\left(i\right)}& (f,V_1^{\left(i\right)},V_2^{\left(i\right)})=\hfill \\ \hfill & \left({\displaystyle \frac{2\pi n_{eff}{L}_2^{\left(i\right)}f}{c}}+{\varphi }_{app,2}^{\left(i\right)}\left(V_2^{\left(i\right)}\right)\right)-\hfill \\ \hfill & \left({\displaystyle \frac{2\pi n_{eff}{L}_1^{\left(i\right)}f}{c}}+{\varphi }_{app,1}^{\left(i\right)}\left(V_1^{\left(i\right)}\right)\right)+{\varphi }_{offset}^{\left(i\right)}\hfill \end{array}$$

(22)

This can be rearranged as:

$$\begin{array}{cc}\hfill {\Phi }^{\left(i\right)}& (f,V_1^{\left(i\right)},V_2^{\left(i\right)})=\hfill \\ \hfill & {\displaystyle \frac{2\pi n_{eff}(L_2^{\left(i\right)}-L_1^{\left(i\right)})f}{c}}+\hfill \\ \hfill & \left({\varphi }_{\mathrm{app},2}^{\left(i\right)}\left(V_2^{\left(i\right)}\right)-{\varphi }_{\mathrm{app},1}^{\left(i\right)}\left(V_1^{\left(i\right)}\right)\right)+{\varphi }_{\mathrm{offset}}^{\left(i\right)}\hfill \end{array}$$

(23)

Let $\Delta L^{\left(i\right)}=L_2^{\left(i\right)}-L_1^{\left(i\right)}$ be the physical path length difference, and let $\Delta {\varphi }_{app}^{\left(i\right)}(V_1^{\left(i\right)},V_2^{\left(i\right)})={\alpha }^{\prime }{L}_{mod}^{\left(i\right)}(V_2^{\left(i\right)}-V_1^{\left(i\right)})$ be the differential applied phase shift. Then, the total phase difference is:

$$\begin{array}{cc}\hfill {\Phi }^{\left(i\right)}& (f,V_1^{\left(i\right)},V_2^{\left(i\right)})=\hfill \\ \hfill & {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}f}{c}}+\hfill \\ \hfill & \Delta {\varphi }_{app}^{\left(i\right)}(V_1^{\left(i\right)},V_2^{\left(i\right)})+{\varphi }_{offset}^{\left(i\right)}\hfill \end{array}$$

(24)

4.2. Frequency-Voltage Relationship

Let $f_{designed}^{\left(i\right)}$ be the central frequency of the MZI stage i when no differential tuning voltage is applied (i.e., $\Delta {\varphi }_{app}^{\left(i\right)}=0$, implying $V_1^{\left(i\right)}=V_2^{\left(i\right)}$ or fixed biases). At this frequency, the resonance condition is:

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{designed}^{\left(i\right)}}{c}}+{\varphi }_{offset}^{\left(i\right)}=2k_0\pi \end{array}$$

(25)

where $k_0$ is an integer representing the initial resonant mode for this MZI stage.

Our objective is to shift the resonant peak to a new central frequency $f_{new}^{\left(i\right)}$ by applying appropriate voltages $V_1^{\left(i\right)}$ and $V_2^{\left(i\right)}$. The new resonance condition at $f_{new}^{\left(i\right)}$ must also be an integer multiple of $2\pi$:

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{new}^{\left(i\right)}}{c}}+{\varphi }_{offset}^{\left(i\right)}+\Delta {\varphi }_{app,tune}^{\left(i\right)}=2k_0\pi \end{array}$$

(26)

Equating the expressions for $2k_0\pi$ from both cases:

$$\begin{array}{cc}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{new}^{\left(i\right)}}{c}}& +{\varphi }_{offset}^{\left(i\right)}+\Delta {\varphi }_{app,tune}^{\left(i\right)}=\hfill \\ \hfill & {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{designed}^{\left(i\right)}}{c}}+{\varphi }_{offset}^{\left(i\right)}\hfill \end{array}$$

(27)

The initial phase offset ${\varphi }_{offset}^{\left(i\right)}$ cancels out:

$$\begin{array}{cc}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{new}^{\left(i\right)}}{c}}+\Delta {\varphi }_{app,tune}^{\left(i\right)}& =\hfill \\ \hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{designed}^{\left(i\right)}}{c}}\end{array}$$

(28)

Solving for the required differential tuning phase shift $\Delta {\varphi }_{app,tune}^{\left(i\right)}$:

$$\begin{array}{c}\hfill \Delta {\varphi }_{app,tune}^{\left(i\right)}={\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}(f_{designed}^{\left(i\right)}-f_{new}^{\left(i\right)})}{c}}\end{array}$$

(29)

Now, substituting $\Delta {\varphi }_{app,tune}^{\left(i\right)}={\alpha }^{\prime }{L}_{mod}^{\left(i\right)}(V_2^{\left(i\right)}-V_1^{\left(i\right)})$:

$$\begin{array}{cc}\hfill {\alpha }^{\prime }{L}_{mod}^{\left(i\right)}& (V_2^{\left(i\right)}-V_1^{\left(i\right)})=\hfill \\ \hfill & {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}(f_{designed}^{\left(i\right)}-f_{new}^{\left(i\right)})}{c}}\hfill \end{array}$$

(30)

Let $V_{diff}^{\left(i\right)}=V_2^{\left(i\right)}-V_1^{\left(i\right)}$ be the required differential tuning voltage for stage i. Then:

$$\begin{array}{c}\hfill V_{diff}^{\left(i\right)}={\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}(f_{designed}^{\left(i\right)}-f_{new}^{\left(i\right)})}{c{\alpha }^{\prime }{L}_{mod}^{\left(i\right)}}}\end{array}$$

(31)

where ${\alpha }^{\prime }$ is the phase change in radians per V·m, representing the electro-optic efficiency.

This equation clearly demonstrates a linear relationship (Figure 7 and Figure 8) between the required differential voltage $V_{diff}^{\left(i\right)}$ and the desired frequency shift $(f_{designed}^{\left(i\right)}-f_{new}^{\left(i\right)})$. The slope of this relationship, $K^{\left(i\right)}={\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}}{c{\alpha }^{\prime }{L}_{mod}^{\left(i\right)}}}$, is a constant for each MZI stage. This derivation confirms that by applying a specific differential voltage, the resonant peak can be linearly shifted.

4.3. Voltage Dependence of Free Spectral Range During Tuning

The FSR of an MZI is the frequency separation between consecutive resonant peaks. A transmission maximum occurs when the total phase difference, ${\Phi }^{\left(i\right)}$, between the MZI arms is an integer multiple of $2\pi$.

For MZI stage i, the total phase difference is:

$$\begin{array}{cc}\hfill {\Phi }^{\left(i\right)}& (f,V_1^{\left(i\right)},V_2^{\left(i\right)})=\hfill \\ \hfill & {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}f}{c}}+{\varphi }_{offset}^{\left(i\right)}+\Delta {\varphi }_{app}^{\left(i\right)}(V_1^{\left(i\right)},V_2^{\left(i\right)})\hfill \end{array}$$

(32)

where f is the optical frequency, $\Delta L^{\left(i\right)}=L_2^{\left(i\right)}-L_1^{\left(i\right)}$ is the fixed physical path length difference, ${\varphi }_{offset}^{\left(i\right)}$ is a fixed initial phase offset, and $\Delta {\varphi }_{app}^{\left(i\right)}$ is the applied differential phase shift due to tuning voltages.

Let $f_m$ and $f_{m+1}$ be the frequencies of two adjacent resonant peaks. The resonance conditions are:

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_m}{c}}+{\varphi }_{offset}^{\left(i\right)}+\Delta {\varphi }_{app}^{\left(i\right)}=2m\pi \end{array}$$

(33)

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{m+1}}{c}}+{\varphi }_{offset}^{\left(i\right)}+\Delta {\varphi }_{app}^{\left(i\right)}=2(m+1)\pi \end{array}$$

(34)

To find the frequency difference, $FS{R}^{\left(i\right)}=f_{m+1}-f_m$, we subtract Equation (33) from Equation (34):

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_{m+1}}{c}}-{\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}{f}_m}{c}}\right)+& \\ \hfill ({\varphi }_{offset}^{\left(i\right)}-{\varphi }_{offset}^{\left(i\right)})+(\Delta {\varphi }_{app}^{\left(i\right)}-\Delta {\varphi }_{app}^{\left(i\right)})=& \\ \hfill 2(m+1)\pi -2m\pi \end{array}$$

(35)

Both the fixed initial phase offset (${\varphi }_{offset}^{\left(i\right)}$) and the applied differential phase shift ($\Delta {\varphi }_{app}^{\left(i\right)}$) terms cancel out. This occurs because these terms affect both resonant peaks equally, resulting in a uniform shift of the entire spectrum.

The equation simplifies to:

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi n_{eff}\Delta L^{\left(i\right)}}{c}}(f_{m+1}-f_m)=2\pi \end{array}$$

(36)

If $FS{R}^{\left(i\right)}=f_{m+1}-f_m$, then:

$$\begin{array}{c}\hfill FS{R}^{\left(i\right)}={\displaystyle \frac{c}{n_{eff}\Delta L^{\left(i\right)}}}\end{array}$$

(37)

Then, for this configuration, the FSR is only determined by the speed of light, the effective refractive index, and the fixed physical path length difference ($\Delta L^{\left(i\right)}$) and does not change with the applied voltage: the voltage merely shifts the position of the resonant peaks, not their spacing.

4.4. Phase Change Dependency on the Electro-Optic Coefficient

For an MZI equipped with an electro-optical phase modulator, the induced phase shift ${\varphi }_{app}$ is a function of the applied voltage V. Assuming a linear electro-optic effect (e.g., Pockels effect) and a modulator with an effective interaction length $L_{mod}$, the phase shift can be expressed as:

$$\begin{array}{c}\hfill {\varphi }_{app}\left(V\right)={\displaystyle \frac{2\pi n_{eff}^3{r}_{33}}{{\lambda }_0}}V{L}_{mod}\end{array}$$

(38)

where $r_{33}$ is the electro-optic coefficient, and ${\lambda }_0$ is the vacuum wavelength. It relates to the electro-optical efficiency constant ${\alpha }^{\prime }$ as:

$$\begin{array}{c}\hfill {\alpha }^{\prime }={\displaystyle \frac{2\pi n_{eff}^3{r}_{33}}{{\lambda }_0}}\end{array}$$

(39)

For the EO, the overlap factor ($\Gamma$) between the optical mode and the applied electric field is a critical parameter. In this analysis, we assume an idealized scenario where $\Gamma$ is considered constant across the entire operational frequency and tuning range. Should this assumption not hold, a more rigorous calculation detailing the frequency-dependent modal overlap for the specific modulators involved would be required.

5. Continuous Resonant Peak Tuning Circuit

To continuously tune the resonant peak of an MZI-based filter, a common approach involves applying a precise voltage to its phase modulators. Here, we propose a simplified electronic circuit that uses a single input current source and a series of impedances (resistors) to generate the required differential tuning voltages for multiple cascaded MZI stages. This design emphasizes simplicity and linear control.

5.1. Circuit Design and Operation

Consider a cascaded MZI filter comprising N stages, where each stage i requires a specific differential tuning voltage $V_{diff}^{\left(i\right)}$ to maintain the desired filter characteristic at a shifted central frequency. In Section 4, we establish that $V_{diff}^{\left(i\right)}$ is linearly proportional to the desired frequency shift $\Delta f=f_{new}-f_{designed}$:

$$\begin{array}{c}\hfill V_{diff}^{\left(i\right)}=C_i\Delta f\end{array}$$

(40)

where $C_i={\displaystyle \frac{2\pi n_g\Delta L^{\left(i\right)}}{c{\alpha }^{\prime }{L}_{mod}^{\left(i\right)}}}$ is a stage-specific constant.

The goal is to generate these N differential voltages $V_{diff}^{\left(i\right)}$ from a single control input. A simple method is to use a current source that feeds a circuit of resistors in series configuration to generate the individual voltages for each arm relative to a common reference, allowing the differential voltages to be set. Those resistors can be chosen after the PIC fabrication to match the correct tuning voltage for each stage, correcting possible fabrication inexactitude.

5.1.1. Proposed Circuit Parts (Figure 9)

• A single tunable current source ($I_{tune}$): This current source will be the master control, with its magnitude directly proportional to the desired frequency shift $\Delta f$.
• Series Resistors ($R_1,R_2,\dots ,R_{2N}$): These resistors are connected in series. The voltages across specific segments of the chain will provide the necessary individual voltages for each MZI arm.
• Voltage Taps: Connections are made at specific points along the series resistor chain to apply the voltages $V_1^{\left(i\right)}$ and $V_2^{\left(i\right)}$ to the phase modulators of each MZI stage. A common ground or reference point serves as the $0V$ reference.

Figure 9. Conceptual Circuit Configuration for a Tuning the PIC filter. The tunable current $I_{tune}$ flows through a resistor ladder, generating voltages $V_{MZ{I}_i}$.

Figure 9. Conceptual Circuit Configuration for a Tuning the PIC filter. The tunable current $I_{tune}$ flows through a resistor ladder, generating voltages $V_{MZ{I}_i}$.

For each MZI stage i, we need to ensure $V_2^{\left(i\right)}-V_1^{\left(i\right)}=C_i\Delta f$. Let $I_{tune}=k_I\Delta f$ for some constant $k_I$.

5.1.2. Circuit Operation

Individual voltages $V_1^{\left(i\right)}$ and $V_2^{\left(i\right)}$ can be tapped from the appropriate points in the series resistor chain. For instance, if $V_1^{\left(i\right)}$ is fixed to a common bias $V_{bias}$, then $V_2^{\left(i\right)}$ needs to be $V_{bias}+V_{diff}^{\left(i\right)}$. This can be achieved by having resistors that set $V_{bias}$ for Arm 1, and then an additional resistor for Arm 2 in series to create $V_{diff}^{\left(i\right)}$. The differential voltage for stage i, $V_{diff}^{\left(i\right)}=V_2^{\left(i\right)}-V_1^{\left(i\right)}$, is thus determined by the voltage drop across the resistors between the two taps for that stage.

Let the voltage at tap $T_{1,i}$ be $V_{1,i}$ and at tap $T_{2,i}$ be $V_{2,i}$. These taps are within the resistor chain. The total resistance between $T_{1,i}$ and $T_{2,i}$ is $R_{tap,i}$ ($R_{tap,i}=R_{n,1}+R_{n,2}+\dots +R_{i-1,1}+R_{i-1,2}$). Then $V_{2,i}-V_{1,i}=I_{tune}{R}_{tap,i}$. We need $I_{tune}{R}_{tap,i}=C_i\Delta f$. Since $I_{tune}=k_I\Delta f$, we have $k_I\Delta f{R}_{tap,i}=C_i\Delta f$. This implies $R_{tap,i}=C_i/k_I$.

Thus, for each MZI stage, the resistors between the taps that feed the modulators for arm 1 and arm 2 must be chosen such that their resistance $R_{tap,i}$ is proportional to the stage’s specific tuning constant $C_i$.

Advantages of this circuit:

• Single Control Input: A single tunable current source ($I_{tune}$) controls the entire network’s frequency shift.
• Linearity: Maintains the linear relationship between the input control (current) and the resulting frequency shift.
• Simplicity: Uses only a current source and passive resistors, minimizing circuit complexity.
• Scalability: Can be extended to an arbitrary number of MZI stages by adding more series resistor segments and corresponding voltage taps.

This circuit effectively transforms a single current input into a set of proportionally scaled differential voltages, precisely tuning each MZI stage to achieve a continuous shift of the overall filter’s resonant peak while maintaining the designed FSR.

6. Feasibility of the Design

Let the differential tuning voltage be denoted as $V_{diff}^{\left(i\right)}=V_2^{\left(i\right)}-V_1^{\left(i\right)}$, for stage i. To ascertain the manufacturing specifications essential for implementing the designed filter, the relationship governing the tuning frequency (Equation (31)) is utilized. For a singular stage, this relationship can be algebraically manipulated to yield:

$$\begin{array}{c}\hfill f_{new}=f_{designed}-{\displaystyle \frac{V_{diff}c{\alpha }^{\prime }{L}_{mod}}{2\pi n_{eff}\Delta L}}\end{array}$$

(41)

To determine the sensitivity of $f_{new}$ to changes in each physical parameter, we compute the partial derivative of $f_{new}$ with respect to that parameter.

6.1. Sensitivity to Differential Tuning Voltage ($V_{diff}$)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{new}}{\partial V_{diff}}}& =-{\displaystyle \frac{c{\alpha }^{\prime }{L}_{mod}}{2\pi n_{eff}\Delta L}}\hfill \end{array}$$

(42)

6.2. Sensitivity to Electro-Optic Efficiency (${\alpha }^{\prime }$)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{new}}{\partial {\alpha }^{\prime }}}& =-{\displaystyle \frac{V_{diff}c{L}_{mod}}{2\pi n_{eff}\Delta L}}\hfill \end{array}$$

(43)

6.3. Sensitivity to Modulator Length ($L_{mod}$)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{new}}{\partial L_{mod}}}& =-{\displaystyle \frac{V_{diff}c{\alpha }^{\prime }}{2\pi n_{eff}\Delta L}}\hfill \end{array}$$

(44)

6.4. Sensitivity to ($n_{eff}$)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{new}}{\partial n_{eff}}}& ={\displaystyle \frac{V_{diff}c{\alpha }^{\prime }{L}_{mod}}{2\pi n_{eff}^2\Delta L}}\hfill \end{array}$$

(45)

6.5. Sensitivity to Optical Path Length Difference ($\Delta L$)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial f_{new}}{\partial \Delta L}}& ={\displaystyle \frac{V_{diff}c{\alpha }^{\prime }{L}_{mod}}{2\pi n_{eff}{(\Delta L)}^2}}\hfill \end{array}$$

(46)

6.6. Error Accounting

To maintain the relative error of $f_{new}$ below 100·E%, we require:

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{\Delta f_{new}}{f_{new}}}\right|<E\end{array}$$

(47)

If there is an independent and identically distributed error for each of the physical variables, the Root Sum Squared (RSS) Tolerance Analysis is useful to calculate the total error. The total change in $f_{new}$ ($\Delta f_{new}$) is approximated by the total differential. Then, to calculate error propagation, the variance formula for $f_{new}$ due to independent errors in the parameters $X_i$ is:

$$\begin{array}{c}\hfill \Delta f_{new}^{\mathrm{RSS}}=\sqrt{\sum _i{\left({\displaystyle \frac{\partial f_{new}}{\partial X_i}}\Delta X_i\right)}^2}\end{array}$$

(48)

Then the RSS error is given by:

$$\begin{array}{cc}\hfill {\left(\Delta f_{new}^{\mathrm{RSS}}\right)}^2& =\hfill \\ \hfill & {\left({\displaystyle \frac{\partial f_{new}}{\partial V_{diff}}}\Delta V_{diff}\right)}^2+{\left({\displaystyle \frac{\partial f_{new}}{\partial {\alpha }^{\prime }}}\Delta {\alpha }^{\prime }\right)}^2+\hfill \\ \hfill & {\left({\displaystyle \frac{\partial f_{new}}{\partial L_{mod}}}\Delta L_{mod}\right)}^2+{\left({\displaystyle \frac{\partial f_{new}}{\partial n_{eff}}}\Delta n_{eff}\right)}^2+\hfill \\ \hfill & {\left({\displaystyle \frac{\partial f_{new}}{\partial \Delta L}}\Delta (\Delta L)\right)}^2\hfill \end{array}$$

(49)

We need $\Delta f_{new}^{\mathrm{RSS}}<E\left|f_{new}\right|$. Let $E_{\mathrm{total}}=E\left|f_{new}\right|$. Assuming that each of the N parameters contributes equally to the squared error, the maximum allowed absolute error for each parameter $X_i$ is given by:

$$\begin{array}{c}\hfill |\Delta X_i| <{\displaystyle \frac{E_{\mathrm{total}}}{\sqrt{N}\left|\frac{\partial f_{new}}{\partial X_i}\right|}}\end{array}$$

(50)

In this analysis, we have $N=5$ physical parameters.

6.6.1. Maximum Tolerated Error for $V_{diff}$

$$\begin{array}{c}\hfill |\Delta V_{diff}| <{\displaystyle \frac{E|f_{new}|2\pi n_{eff}\Delta L}{\sqrt{6}c{\alpha }^{\prime }{L}_{mod}}}\end{array}$$

(51)

6.6.2. Maximum Tolerated Error for ${\alpha }^{\prime }$

$$\begin{array}{c}\hfill |\Delta {\alpha }^{\prime }| <{\displaystyle \frac{E|f_{new}|2\pi n_{eff}\Delta L}{\sqrt{6}{V}_{diff}c{L}_{mod}}}\end{array}$$

(52)

6.6.3. Maximum Tolerated Error for $L_{mod}$

$$\begin{array}{c}\hfill |\Delta L_{mod}| <{\displaystyle \frac{E|f_{new}|2\pi n_{eff}\Delta L}{\sqrt{6}{V}_{diff}c{\alpha }^{\prime }}}\end{array}$$

(53)

6.6.4. Maximum Tolerated Error for $n_{eff}$

$$\begin{array}{c}\hfill |\Delta n_{eff}| <{\displaystyle \frac{E|f_{new}|2\pi n_{eff}^2\Delta L}{\sqrt{6}{V}_{diff}c{\alpha }^{\prime }{L}_{mod}}}\end{array}$$

(54)

6.6.5. Maximum Tolerated Error for $\Delta L$

$$\begin{array}{c}\hfill |\Delta (\Delta L)|<{\displaystyle \frac{E|f_{new}|2\pi n_{eff}{(\Delta L)}^2}{\sqrt{6}{V}_{diff}c{\alpha }^{\prime }{L}_{mod}}}\end{array}$$

(55)

If the error in $f_{new}$ is set as $|\Delta f_{new}| <1 \mathrm{MHz}$. Thus, the error limit for $f_{new}$ is $\Delta f_{new}^{\mathrm{limit}}=1 \mathrm{MHz}$ (E $\sim 5.2\times {10}^{-9}$).

The root-sum-square (RSS) error propagation method distributes this total error limit among the individual physical parameters. To ensure $\Delta f_{new}^{\mathrm{RSS}}<\Delta f_{new}^{\mathrm{limit}}$, and assuming an equal distribution of the squared error budget among the N parameters, the maximum tolerated error for each parameter $\Delta X_i$ is:

$$\begin{array}{c}\hfill |\Delta X_i| <{\displaystyle \frac{\Delta f_{new}^{\mathrm{limit}}}{\sqrt{N}\left|\frac{\partial f_{new}}{\partial X_i}\right|}}\end{array}$$

(56)

Here, $N=5$ parameters are considered: $V_{diff},{\alpha }^{\prime },L_{mod},n_{eff},\Delta L$.

6.7. Example System Parameters

The following nominal values for the physical parameters (similar as Lithium niobate) are used in the calculation of the RRS error:

• $f_{new}=193.0\times {10}^{12} \mathrm{Hz}$.
• $n_{eff}=2.500$.
• $\Delta L=1.000\times {10}^{-3} \mathrm{m}$.
• $L_{mod}=1.000\times {10}^{-3} \mathrm{m}$.
• ${\alpha }^{\prime }=\pi /3\times {10}^{-3} \mathrm{V}\mathrm{m}\approx 1047 \mathrm{rad}{\mathrm{V}}^{-1}\mathrm{m}$.
• $V_{diff}=1.000 \mathrm{V}$.
• $c\approx 2.99792458\times {10}^8 {\mathrm{m}\mathrm{s}}^{-1}$.

The target absolute error limit for $f_{new}$ is $\Delta f_{new}^{\mathrm{limit}}=1.000\times {10}^6\mathrm{Hz}$.

The values in

Table 1. Maximum Tolerated Errors for System Parameters (for $\Delta f_{new}<1\mathrm{MHz}$).

Variable	Maximum Tolerated Error	SI Unit
$V_{diff}$	$2.038\times {10}^{-5}$	$\mathrm{V}$
${\alpha }^{\prime }$	$2.135\times {10}^{-2}$	$\mathrm{rad}{\mathrm{V}}^{-1}\mathrm{m}$
$L_{mod}$	$2.038\times {10}^{-8}$	$\mathrm{m}$
$n_{eff}$	$5.096\times {10}^{-5}$	-
$\Delta L$	$2.038\times {10}^{-8}$	$\mathrm{m}$

indicate the sensitivity of the tuning frequency to variations in the parameters for a $1\mathrm{MHz}$ absolute error limit on $f_{new}$.

7. Conclusions

This paper has presented the theoretical foundation for designing and tuning advanced band-pass filters utilizing cascaded MZI networks. We demonstrated that a band-pass filter with a specified width (FSR₀) and an arbitrarily large effective FSR can be generated with a minimum number of MZIs by selecting their individual FSRs as prime multiples of FSR₀. The total power transmission for such cascaded filters as well as the conditions for the preservation of the MZI’s intrinsic FSR during the tuning process was deduced. A simplified electronic circuit was proposed, utilizing a single input current and series resistors, to continuously tune the resonant peak across multiple MZI stages, highlighting a practical approach to reconfigurable filter implementation.