SECTION 1

Microwave photonic filter (MPF) has a lot of potential applications in dealing with radio frequency (RF) signals. By taking the advantages of photonic techniques, it is able to implement flexible filtering functions with low loss and high bandwidth, as well as immunity to electromagnetic interference (EMI), tunability, and reconfigurability [1], [2], [3]. The general concept of an MPF is a digital filter technology. The RF input signal is modulated onto one or multiple optical carriers via an electrical-optical modulator, and then, the modulated optical signals are fed to a photonic system that samples, weighs, and combines them by using optical delay lines or other photonic elements. When the signals are recovered at a photodetector (PD), the converted electrical signal is modified corresponding to the transfer function of the photonic system [3], [4].

In general, the incoherent light sources are employed to implement a stable MPF since the phase fluctuation will have no much influence on the performance of an MPF and consequently against environmental changes. There are two main approaches to obtain the incoherent optical taps. One is to use multiwavelength lasers, an array of independent lasers, or an optical frequency comb [5], [6], [7]; the other is to use a wide-band optical source (WBOS) sliced by a periodic or programmable multichannel optical filter [8], [9], [10]. In all cases, each source implements a filter sample, and the sampling period is determined by dispersive media.

For the MPFs based on multiwavelength lasers, laser arrays, or optical frequency comb, the main drawbacks are related to the high cost of their configuration and the electrical filtering response with an undesired periodic spectral characteristic, which impose limitations in some specific applications. A cost-effective way to realize single-bandpass MPFs is using a sliced WBOS. In this case, the taps are not a discrete distribution as the former case, then the free spectral range (FSR) goes to infinity, which is able to contribute a single-band response [9], [10].

Although a single-passband MPF is very important for some photonic microwave applications, e.g., an optoelectronic oscillator [10], one fact is that, in modern wireless and satellite systems, a bandpass filter with multiple passbands is considered as a key component, which is neither periodic nor single RF response and can simultaneously transmit multiple desired noncontiguous channels [11]. As far as we know, MPFs with such response have not been reported.

In this paper, we propose a novel multiband bandpass MPF, which has selectable one or more passband responses. The schematic of the proposed MPF is illustrated in Fig. 1. A WBOS is sliced by an optical filter to generate sampling taps, where a high-birefringence fiber loop mirror (HB-FLM) with two segments of high-birefringence fiber (HBF) is employed. Corresponding to the polarization rotation of the light in the loop, the HB-FLM shows various periodical spectral characteristics, which lead to uniform or multiple periods on the transmission spectrum. After the multiwavelength source modulated by a phase modulator (PM) and delayed by dispersive medium, the RF response exhibits one or multiple passbands in accordance with the sampling periods. In addition, the response on baseband is suppressed.

SECTION 2

In order to simplify the formula derivation and make comparisons, let us briefly review a single-passband MPF based on a WBOS sliced by a Mach–Zehnder interferometer (MZI) first, which has been analyzed and demonstrated in [9] and [12]. For a typical MZI, it consists of two ideal 3-dB fiber couplers with the arm lengths of ${\rm L}_{\rm up}$ and ${\rm L}_{\rm down}$, as shown in Fig. 2. When the light field is launched into port 1 and output from port 3, the transmittivity is given by TeX Source $$T_{MZI} (\omega) = {1 \over 2}\left[1 - \cos \phi (\omega)\right].\eqno{\hbox{(1)}}$$

Here, $\omega$ is optical frequency, and $\phi (\omega)$ is the phase difference between the two arms, which can be calculated as TeX Source $$\phi (\omega) = \beta_{0} (L_{up} - L_{down}) = \beta_{0}\Delta L\eqno{\hbox{(2)}}$$ where $\beta_{0}$ is the propagation constant, and the wavelength spacing (FSR of the optical spectrum) is TeX Source $$FSR_{MZI} = 2\pi c/n\Delta L$$ where $c$ and $n$ are the light velocity and effective index of fiber. When a WBOS is sliced by this MZI, an approximately cosinoidal transmission spectrum can be obtained. There is only one FSR is presented for a certain $\Delta L$. If the multiwavelength light from the slicing filter is modulated by a PM with modulation indexes of $m_{1}$ and $m_{2}$ for the two first-order modulation sidebands and pass through a dispersive medium with the transfer function of $H_{disp}(\omega) = \vert H_{disp}(\omega)\vert \exp [-j\Phi (\omega)]$, the overall system response is TeX Source $$H_{RF} (f) = \int T_{MZI}(\omega) \left[m_{1}H_{disp}^{\ast}(\omega) H_{disp}(\omega - 2\pi f) - m_{2}H_{disp}(\omega)H_{disp}^{\ast}(\omega + 2\pi f) \right]d\omega\eqno{\hbox{(3)}}$$ where $f$ is RF signal frequency. Due to the dispersive medium acting as a phase filter, we can assume $\vert H_{disp}(\omega)\vert = 1$, and the phase dependence can be given by means of a Taylor expansion centered at $\omega_{0}$, as shown in TeX Source $$\Phi (\omega) = \Phi (\omega_{0}) + \tau_{d}(\omega_{0}) (\omega - \omega_{0}) + {1 \over 2}\beta L(\omega - \omega_{0})^{2} + {1 \over 3}\chi L(\omega - \omega_{0})^{3}\eqno{\hbox{(4)}}$$ where $\tau_{d}$ is the group delay, $\beta$ is the dispersion, $\chi$ is the dispersion slope, and $L$ is the length of the dispersive medium. Then, the frequency response of the MPF can be derived as [9], [12] TeX Source $$\eqalign{H_{RF} (f) = &\, \exp \left\{- j \left[2\pi f\tau_{d} (\omega_{0}) + (\chi L/3) (2\pi f)^{3}\right]\right\}\cr\noalign{\vskip-1pt} & \times \bigg\{m_{1}\exp \left[- j (\beta L/2) (2\pi f)^{2}\right] \times \int T_{MZI} (\omega) \cr\noalign{\vskip-1pt} & \qquad\times \exp \left \{- j \left[(2\pi \beta f + 4\pi^{2}\chi f^{2}) (\omega - \omega_{0}) + 2\pi \chi (\omega - \omega_{0})^{2}f\right] L\right\} d\omega\cr\noalign{\vskip-1pt} & \qquad - m_{2}\exp \left[+ j (\beta L/2) (2\pi f)^{2}\right] \times \int T_{MZI} (\omega) \cr\noalign{\vskip-1pt} & \qquad \times \exp \left\{- j \left[(2\pi \beta f - 4\pi^{2}\chi f^{2}) (\omega - \omega_{0}) + 2\pi \chi (\omega - \omega_{0})^{2}f\right] L\right\} d\omega\bigg\}.}$$

The equation above gives a response of single RF passband, and the baseband response is suppressed. The center frequency $f_{c}$ of the MPF is determined by the delay time of the dispersive media as [9], [13] TeX Source $$f_{c} = {1/\left[ \beta L(FSR_{MZI})\right]}.\eqno{\hbox{(5)}}$$

In addition, if the effect of the dispersion slope within the optical spectral bandwidth can be neglected, the 3-dB bandwidth of the RF passband is independent of $f_{c}$ and can be expressed as [9] TeX Source $$\Delta f_{c} = {\sqrt{8\ln 2}/\left(\beta L \cdot \Delta \omega \right)}\eqno{\hbox{(6)}}$$ where $\Delta \omega$ is the optical source bandwidth.

Different from the MZI-based MPFs, a two-section HB-FLM, which can also be referred to as a two-order Solc-type filter, is employed as a slicing filter in our proposed MPF. As shown in Fig. 3, the incident light at the input port (port 1) is split by the coupler into two beams with clockwise (CW) and counter-CW (CCW) propagation directions, respectively. The two beams propagate along the loop and then recombine and interfere at the coupler with different phases due to the birefringence of the HBFs. In the loop, the two segments of HBF have lengths of $L_{1}$ and $L_{2}$, and the refractive indexes of the fast and slow axes are $n_{f}$ and $n_{s}$. $\theta_{i}\ (i = 1, 2, 3, 4)$ denotes the angle between the fast axis of the HBFs and the $x$-axis of the local coordinate at the incident plane. For a coupler with the power-splitting ratio of $k$, the E-fields at ports 3 and 4 can be given by TeX Source $$\left[\matrix{E_{3}\cr E_{4}}\right] = \left[\matrix{\sqrt{1 - k} & j\sqrt{k}\cr j\sqrt{k} & \sqrt{1 - k}} \right]\left[\matrix{E_{1in}\cr 0}\right]\eqno{\hbox{(7)}}$$ where $E_{1in}$ is the input light field at port 1.

Now, let us consider the Jones transfer matrices of this HB-FLM. Assuming that the two segments of the HBF have the same beat length, the matrices along the two directions are given by [14] TeX Source $$\eqalignno{M_{cw} = &\, R (\theta_{3}) J_{2}R (\theta_{2}) J_{1}R (- \theta_{1})&\hbox{(8a)}\cr M_{ccw} = &\, R (\theta_{1}) J_{1}R (- \theta_{2}) J_{2}R (-\theta_{3})&\hbox{(8b)}}$$ where $R(\theta_{i}) (i = 1, 2)$ denotes the rotation matrix of the polarization state induced by the HBF with the incident angle $\theta_{i}$, and $J_{i}$ is the phase delay matrix caused by the HBFs. The expressions of $R (\theta_{i})$ and $J_{i}$ are TeX Source $$\eqalignno{R (\theta_{i}) = &\, \left[\matrix{\cos \theta_{i} & - \sin \theta_{i}\cr\noalign{\vskip-1pt} \sin \theta_{i} & \cos \theta_{i}}\right]&\hbox{(9a)}\cr\noalign{\vskip-1pt} J_{i} = &\, \left[\matrix{\exp (- j2\pi n_{f}L_{i}/ \lambda) & 0\cr\noalign{\vskip-1pt} 0 & \exp (- j2\pi n_{s}L_{i}/\lambda)}\right]&\hbox{(9b)}}$$ where $L_{i}\ (i = 1, 2)$ are the lengths of the HBFs used in the Sagnac loop. Then, the transmittivity can be expressed as TeX Source $$\displaylines{T_{FLM}(\omega) = (1 - 2k)^{2} + 4k (1 - k) \left[\sin \left(\theta_{1} - \theta_{3}\right)\cos \theta_{2}\cos \left({\pi \Delta n (L_{1} + L_{2}) \over \lambda}\right)\right.\hfill\cr\noalign{\vskip8pt}\hfill \left. +\ \cos \left(\theta_{1} - \theta_{3}\right)\sin \theta_{2}\cos \left({\pi \Delta n (L_{1} - L_{2}) \over \lambda}\right) \right]^{2}.\quad\hbox{(10)}}$$

Here, we have $\Delta n = \vert n_{f} - n_{s} \vert$. For a 3-dB coupler $(k = 0.5)$, the transmittivity is simplified as TeX Source $$\eqalignno{T_{FLM}(\omega) = &\, \left\{A\cos \left[{\phi_{1}(\omega) + \phi_{2}(\omega) \over 2}\right] + B\cos \left[{\phi_{1}(\omega) - \phi_{2}(\omega) \over 2}\right] \right \}^{2}\cr\noalign{\vskip8pt} = &\, \left({A^{2} \over 2} + {B^{2} \over 2}\right) + {A^{2} \over 2}\cos \left[ \phi_{1}(\omega) + \phi_{2}(\omega)\right] + AB\cos \phi_{1}(\omega) + AB\cos \phi_{2}(\omega) \cr\noalign{\vskip8pt} & + {B^{2} \over 2}\cos \left[\phi_{1}(\omega) - \phi_{2}(\omega)\right]&\hbox{(11)}}$$ where $A = \sin (\theta_{1} - \theta_{3}) \cos \theta_{2}$, $B = \cos (\theta_{1} - \theta_{3}) \sin \theta_{2}$, $\phi_{1}(\omega) = 2\pi \Delta nL_{1}/\lambda$, and $\phi_{2}(\omega) = 2\pi \Delta nL_{2}/\lambda$ are defined. Angles $\theta = \theta_{1} - \theta_{3}$ and $\theta_{2}$ can be controlled and varied by the two polarization controllers (PCs) in the loop. According to (11), $T_{FLM}(\omega)$ includes four types of wavelength spacing, which are determined by $\lambda^{2}/\Delta nL_{1}$, $\lambda^{2}/\Delta nL_{2}$, $\lambda^{2}/\Delta n (L_{1} + L_{2})$, and $\lambda^{2}/\Delta n (L_{1} - L_{2})$, respectively.

Assuming the two HBFs have $L_{1} = 2L_{2}$, hence, $\phi_{1}(\omega) = 2\phi_{2}(\omega)$. $T_{FLM}(\omega)$ can be deduced as TeX Source $$\eqalignno{T_{FLM}(\omega) = &\, \left({A^{2} \over 2} + {B^{2} \over 2}\right) + {A^{2} \over 2}\cos 3\phi_{2}(\omega) + AB\cos 2\phi_{2}(\omega) + \left(AB + {B^{2} \over 2}\right)\cos \phi_{2}(\omega)\cr\noalign{\vskip8pt} = &\, - 2AB + {A^{2} \over 2}\left[1 + \cos 3\phi_{2}(\omega)\right] + AB\left[1 + \cos 2\phi_{2}(\omega)\right] + \left(AB + {B^{2} \over 2}\right)\left[1 + \cos \phi_{2}(\omega)\right]\cr\noalign{\vskip8pt} = &\, - 2AB + T_{MZI, 1}(\omega) + T_{MZI, 2}(\omega) + T_{MZI, 3}(\omega).&\hbox{(12)}}$$

As shown in (12), $T_{FLM}(\omega)$ is a superimposition of three cosine functions with different periods. It can also be found that the transmission spectrum of the two-order HB-FLM is equivalent to three parallel (not cascaded) MZI slicers with coupled and modified weight coefficients as compared to (1). The corresponding periods are $\lambda^{2}/3\Delta nL_{2}$, $\lambda^{2}/2\Delta nL_{2}$, and $\lambda^{2}/\Delta nL_{2}$, respectively. The minimum period is $\lambda^{2}/3\Delta nL_{2}$, which is referred to the fundamental FSR and determined by $L_{1} + L_{2}$. When $\theta$ and $\theta_{2}$ are adjusted, the contribution of each term is changed and the dominant FSR is varied consequently. For example, if the conditions of $\theta = 0.5\pi$ and $\theta_{2} = 0$ are fulfilled, we will have $A = 1$ and $B = 0$. $T_{FLM}(\omega)$ can be rewritten as TeX Source $$T_{FLM}(\omega) = {1 \over 2} \left[1 + \cos 3\phi_{2}(\omega)\right]$$ which is equivalent to the case of one MZI with spectral period of $\lambda^{2}/3\Delta nL_{2}$. For another example, if $\theta = -0.25\pi$ and $\theta_{2} = 0.375\pi$ are selected, we have $\vert AB \vert : \vert A^{2}/2 \vert : \vert AB + B^{2}/2 \vert = 4.83: 1: 1$, which means that $\lambda^{2}/2\Delta nL_{2}$ is the dominant spectral period in $T_{FLM}(\omega)$. Fig. 4 shows the numerically calculated transmittivity. Here, $L_{1}$, $L_{2}$, and $\Delta n$ are set to be 6 m, 3 m, and 0.000385, which has a corresponding beat length of 4 mm. From Fig. 4, the dominant wavelength spacing shows fundamental, doubled, and tripled FSR, when $(\theta, \theta_{2})$ is equal to $(0.5\pi, 0)$, $(-0.25\pi, 0.375\pi)$, and $(0, 0.5\pi)$, respectively. It should be noticed, just like the description above, that the case of 2 × FSR (dot curve) is always mixed with the spectral period of FSR and 3 × FSR; therefore, it has no clear periodical boundary like the case of solid curve and dash curve. However, it can also be distinguished from the comparison with solid curve. Obviously, once other angles are chosen, the sliced spectrum will present various mixing period properties. When this HB-FLM is used as a slicer in an MPF, the corresponding RF frequency response will be TeX Source $$H_{RF} (f) = \int T_{FLM}(\omega) \left[m_{1}H_{disp}^{\ast}(\omega)H_{disp}(\omega - 2\pi f) - m_{2}H_{disp}(\omega)H_{disp}^{\ast}(\omega + 2\pi f)\right]d\omega.\eqno{\hbox{(13)}}$$

Compared with (3), the transfer function includes three possible passband responses, and each one has the same features as the case of MZI. So, by varying $\theta_{i}$, single- or multiple-passband MPF can be realized for adapting different applications.

It should be noticed that even a small length deviation of the HBF can cause great optical spectral change because HBF has a very small beat length, e.g., even 2-mm deviation lead to $\pi$ phase shift on $\phi_{2}(\omega)$. However, the influence on FSR can be ignored. Let us take HBF lengths of 6 m and 2.9991 m for example. Fig. 5 gives the normalized transfer function (13) as a function of the RF frequency $f$, and the insets present the corresponding optical spectrum. Here, we consider that the optical source has Gaussian distribution, the dispersion slope of fiber is null, and $\Delta n$, $\beta$, and $\Delta \omega$ are 0.000385, $-23\ \hbox{ps}^{2}/\hbox{km}$, and 5 THz, respectively. Fig. 5(a)–(c) gives three different situations with uniform optical spectral periods (corresponding to single-band filters), and Fig. 5(d)–(f) show three mixed situations with various periodic combinations (corresponding to multiple passbands filters). From Fig. 5, one can see that the number of the passband is dependent on the dominant optical spectral periods included in the taps. The center frequencies and RF bandwidth are governed by (5) and (6).

Although these results show good passband characteristic, there are still some unwanted peaks especially in Fig. 5(b), (d), and (e). These can be explained by (12). For the case of Fig. 5(b), the dominant term is $AB\cos [2\phi_{2}(\omega)]$, which means that coefficients $A$ and $B$ cannot be zero. What we can do is to balance and optimize the values to make $\vert AB \vert \gg \vert A^{2}/2 \vert$ and $\vert AB \vert \gg \vert AB + B^{2}/2 \vert$. Since $A^{2}/2$ and $AB + B^{2}/2$ are not zero, the relative small contributions of $\cos 3 \phi_{2}(\omega)$ and $\cos \phi_{2}(\omega)$ appeared. Similarly, an unwanted weak responses from $\cos 3\phi_{2}(\omega)$ and $\cos \phi_{2}(\omega)$ also appeared in Fig. 5(d) and (e). As a comparison, in order to eliminate the coupled weight coefficients on each term, the WBOS needs to be split into several paths and sliced by individual MZIs to generate desired spectral period on each path, and then amplified (or attenuated) independently. After recombining these optical fields as sampling taps, a multiband MPF without interaction can be obtained. But this configuration is more complicated and less cost efficient.

SECTION 3

In order to verify the proposed MPF, the experimental demonstration was carried out. The experimental setup is shown in Fig. 6. The used WBOS, whose bandwidth covers the C +L band, is centered at 1570 nm, as shown in inset I. It is first sliced by the two-order HB-FLM to generate taps as shown in the inset II, where two commercial available HBF patchcords with nominal lengths of 6 m and 3 m are used. In order to obtain a quasi-Gaussian-like profile, the sliced source is shaped by an extra HB-FLM with an 8-cm-length HBF inside. When the WBOS directly passes through it, a main lobe centered at 1560 nm with a 3-dB bandwidth about 6 nm is obtained as shown in inset III. The following semiconductor optical amplifier (SOA) also has a quasi-Gaussian-like gain spectrum, which is able to reshape the envelope of the light source approaching Gaussian distribution and compensate the insertion loss caused by the former optical components. Inset IV shows the spontaneous emission spectrum of the SOA. Two isolators (ISOs) are employed to prevent optical reflection. The output of the SOA is modulated by a PM and then delayed by a coil of single mode fiber (SMF), where the fiber lengths of 10 km or 25 km are used for tuning the RF response. After the photodetection by a 12-GHz PD, the corresponding frequency response is characterized by a VNA. By properly aligning the PCs inside the HB-FLM, several typical cases are obtained, whose frequency responses are shown in Fig. 7. The insets are the corresponding sliced optical spectrum with span of 70 nm for the overall spectrum and 5 nm for details, respectively.

From Fig. 7, three available passbands with good out-of-band rejection ratio are clearly seen [e.g., Fig. 7(a)–(c)], and the center frequencies are 3.2 GHz, 2.1 GHz, and 1.05 GHz when using 25-km SMF and 7.78 GHz, 5.21 GHz, and 2.6 GHz when using 10-km SMF, which show a good agreement with the theoretical prediction. As indicated by the theoretical analysis, any of these passbands can be selected to appear alone or as a form of combination provided the desired adjustment of the slicer. For example, two dual-band filters are shown by Fig. 7(d), (e), and (f) gives a triple-band filter.

Comparing the cases of using 25-km and 10-km SMF, the former has a lower power level than the later. The reason is that the taps suffer more transmission loss in longer fiber; meanwhile, the frequency response gets larger suppression when closer to dc due to phase modulation [13]. The width of a RF passband is determined by optical-source bandwidth and the amount of dispersion as described in (4); therefore, for the same optical spectral distribution, the narrower RF bandwidth is observed in the case of using longer SMF. However, under a certain fiber length, one may notice from Fig. 7 that the filter response gives different RF bandwidth on each passband. This is caused by nonnegligible dispersion slop in the SMF [9].

According to the theoretical analysis and experimental demonstration, we can summarize the method for designing a multiband bandpass filter. First, the number of the required passband is determined by the involved periods of the sliced optical spectrum, which is depended on the number of segments of the HBF used and the length relationship among them. Second, the RF response has strict relationship to the wavelength spacing and the accumulated dispersion. A reasonable wavelength spacing combinations with suitable time delay decides a desired passband interval. Finally, under a certain amount of dispersion, the width of RF passbands is determined by the bandwidth of the light source. By using dispersion flattened medium, the bandwidth difference among RF passbands can be eliminated. Based on these factors above, a desired multiband bandpass filter can be obtained by carefully choosing the relevant parameters.

SECTION 4

A novel scheme of a selectable multiband bandpass MPF has been proposed. Theoretical analysis has been carried out to describe the implementation of a two-order HB-FLM, which acted as a spectrum slicer for the optical spectrum. Different from the conventional configuration by using MZI, there are three periods included into the transmission function of the two-order HB-FLM when the length of one HBF is twice of the other. Thanks to its various periodic spectral characteristics, the sliced WBOS provides the taps with single or mixing of different wavelength spacings, which can be considered as the combinations of three MZI slicers with tunable weight. After the time delay and dispersion by SMF, RF response may finally show up to be either three single-band filters, two dual-band filters, or a triple-band filter. Both analytical investigation and experimental demonstration show that the center frequency and width of the RF passbands have the same characteristics as that of MZI-based MPFs but simpler configuration and lower cost.

The proposed multiband MPF shows good flexibility and high spectrum efficiency. If one of the HBFs in the slicer is substituted by a tunable differential group delay line, the periodicity of the optical spectrum can be adjusted. It will induce a tunable multiband bandpass filter consequently. It is worth noticing that, if the original WBOS has a bell-shaped spectrum (i.e., a superluminescent LED) [15], the second HB-FLM and the SOA can be omitted. The MPF configuration has been then further simplified.

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