Abstract

In this paper, a novel microwave photonic filter (MPF) with multiple independently tunable passbands is proposed. A broadband optical source (BOS) is employed and split by a 1:N coupler into several branches. One branch is directed to a phase modulator which is modulated by a radio frequency signal and the other branches are delayed by optical delay lines (ODLs), respectively. All of these branches are combined by another 1:N coupler and sent to a dispersion compensation fiber which is used to introduce group delay dispersion to the optical signal. At a photodetector, each time-delayed broadband lightwave beating with the sidebands produced by the phase modulator forms a passband of the MPF. By tuning the delay of each broadband lightwave, the center frequency of the passband can be independently tuned. An MPF with two independently tunable passbands is experimentally demonstrated. The two passbands can be tuned from DC to 30 GHz with a 3-dB bandwidth of about 250 MHz. The stability and dynamic range of the MPF are also evaluated. By employing more branches delayed by ODLs, more passbands can be generated.

© 2015 Optical Society of America

1. Introduction

Microwave photonic filters (MPFs) have attracted plenty of interest in radio frequency (RF) signal processing as electronic bottleneck can be overcome by the advantages provided by the photonic devices such as large time-bandwidth product, high tunability and reconfigurability, and electronicmagnetic interference (EMI) immunity [1, 2]. Generally, MPFs can be implemented by optical delay line structure with an infinite impulse response (IIR) or a finite impulse response (FIR) [2]. In the frequency domain, MPFs with discrete taps have periodic passbands which cannot be tuned and configured independently. In some applications such as the optoelectronic oscillator [3], MPFs with single tunable passband are required. Previously, these MPFs have been achieved based on stimulated Brillouin scattering (SBS) [4], a ring resonator [5], a phase-shifted fiber Bragg grating (FBG) [6] and a broadband optical source (BOS) [7–9]. On the other hand, recent advances in telecommunication and radar applications demand multifunctional microwave circuits to provide various services in modern RF front-ends [10]. Particularly, multi-band bandpass filters are highly desired in multi-mode transceiver to support multi-standard wireless communications. Microwave filters with dual-, triple-, quad- passband have been proposed to handle different frequency bands [11–13]. However, conventional microwave filters have limited tunability and reconfigurability. In [10–13], typical tuning ranges are within 1 GHz. Following their electrical counterparts, MPFs with multiple passbands have been also proposed recently [14–17]. In [14], a selectable multiband bandpass MPF is achieved by utilizing a BOS and a two-order high-birefringence fiber loop mirror. However, the passbands of the MPF cannot be tuned independently and the tunablility has to be realized by adjusting the differential group delay line which leads to a complicated operation. In [15], a dual-passband MPF is proposed based on a BOS and a modified dual-pass Mach-Zehnder interferometer. Nevertheless, the two passbands of the MPF cannot be tuned independently either. An MPF with two independently tunable passbands is demonstrated by using a phase modulator (PM) and an equivalent phase-shifted FBG in [16]. Compared with the MPF based on a cost-effective BOS, this MPF needs two expensive tunable laser sources. In addition, the tuning ranges of the two passbands are limited within 5.4 GHz and 7.4 GHz and the number of its passbands cannot be extended. By utilizing a phase-shifted FBG combined with polarization modulation and polarization multiplexing, another dual-passband MPF is proposed [17]. However, the tuning range of the MPF is limited within about 6 GHz and the number of passband cannot be extended either.

In this paper, we propose a novel MPF with multiple independently tunable passbands. In our configuration, the broadband lightwave (BL) emitted from a BOS is filtered by an optical filter (OF) and split into several branches by a 1:N coupler. One branch is connected to a PM which is modulated by a RF signal, while other branches are delayed by optical delay lines (ODLs). All of these branches are combined by another 1:N coupler and sent to a dispersion compensation fiber (DCF) which is used to introduce group velocity dispersion (GVD) to the optical signal. After amplified by an erbium-doped fiber amplifier (EDFA), the optical signal is sent to a photodetector (PD) for optical-to-electrical conversion. At the PD, each time-delayed BL beating with the sidebands generated by the PM forms a passband of the MPF. An MPF with two independently tunable passbands is experimentally illustrated. Each passband with a 3-dB bandwidth of about 250 MHz can be independently tuned to 30 GHz. The shapes of the passbands fit well with the simulation results. The stability of the MPF is also investigated. The fluctuation of the center frequency and the magnitude of its 1st passband are within 31 MHz and 0.4 dB and the fluctuation of the center frequency and magnitude of its 2nd passband are within 37 MHz and 0.34 dB, respectively. The SFDRs of the two passbands centered at 1 GHz and 4 GHz are measured to be 73.5 dB·Hz2/3 and 73 dB·Hz2/3, respectively. The number of the MPF’s passbands can be flexibly extended by introducing more branches delayed by ODLs. The bandwidth of the passband can be also reconfigurated by tailoring the OF. We also consider an alternative MPF configuration with multiple passbands. However, the multiple independently tunable passbands of the MPF bring extra passbands, and the carrier suppression effect (CSE) deteriorates the magnitude of the passbands at the same time. Therefore, this MPF with multiple passbands is not suitable for many applications.

2. Principle

The schematic diagram of the proposed MPF is shown in Fig. 1. The amplified spontaneous emission (ASE) emitted from an EDFA is employed as a BOS. An optical filter (OF) is incorporated after the BOS to tailor the spectrum of BL. A 1:N coupler is connected to the OF to split the BL into several branches. One of the branches is connected to a PM followed by a polarization controller (PC). The other branches are connected to ODLs succeeded by variable optical attenuators (VOAs) which are used to adjust the light power of each branch. All of the branches are combined by another 1:N coupler. The output of the 1:N coupler is sent to a dispersion compensation fiber (DCF) module. An EDFA is connected after the DCF to amplify the optical signal. A PD is connected after the EDFA for optical-to-electrical conversion. The RF signal generated by the vector network analyzer (VNA) is applied to the PM and the output of the PD is fed back to the VNA for measuring the frequency response of the MPF. For simplicity, a dual-passband MPF is experimentally illustrated and multiple passbands can be easily formed by adding more time-delayed branches.

 figure: Fig. 1

Fig. 1 Schematic diagram of the proposed MPF with multiple independently tunable passbands. Broadband optical source: BOS, optical filter: OF, optical delay line: ODL, variable optical attenuator: VOA, phase modulator: PM, polarization controller: PC, dispersion compensation fiber: DCF, erbium-doped fiber amplifier: EDFA, photodetector: PD, vector network analyzer: VNA.

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The complex electrical field of the BOS tailored by the OF can be written as

e(t)=12πE(Ω)exp(jΩt)dΩ,
and E(Ω) can be expressed as
E(Ω)=N(Ω)expjθ(Ω),
where N(Ω) is the power spectral density (PSD) and θ(Ω) is the phase of the frequency component. θ(Ω) is a random variable whose statistical properties can be written as [18]
<expjθ(Ω)>=0<expj[θ(Ω)θ(Ω)]>=2πδ(ΩΩ),
where < > denotes ensemble average.

Suppose the PM is modulated by a single-frequency RF signal given by Vecos(ωet) where Ve and ωe are the amplitude and angular frequency of the RF signal, respectively, the electrical field after the PM can be expressed as

e(t)P=kPe(t)exp[jγcos(ωet)]=kPe(t)n=jnJn(γ)exp(jnωet)kPe(t)[J0(γ)+jJ1(γ)exp(jωet)+jJ1(γ)exp(jωet)],
where kp is the power coefficient which can be adjusted by the VOA, γ = πVe/Vπ is the modulation index, Vπ is the half-wave voltage of the PM, Jn denotes the nth-order Bessel function of the first kind. In writing Eq. (4), small-signal modulation is assumed, which means higher-order sidebands are neglected. Applying the Fourier transform to both sides of the Eq. (4), the spectrum of electrical field can be expressed as
EP(Ω)=kPJ0(γ)E(Ω)+jkPJ1(γ)E(Ωωe)+jkPJ1(γ)E(Ω+ωe).
The electrical field of the branch delayed by ODL1 can be expressed as e1(t) = k1(t-t1), where k1 is the power coefficient, and t1 is the time delay compared to the branch of the PM. The spectrum of the electrical field delayed by ODL1 can be written as
E1(Ω)=k1exp(jΩt1)E(Ω).
Similarly, the spectrum of the electrical field delayed by ODL2 can be expressed as
E2(Ω)=k2exp(jΩt2)E(Ω).
The spectrum of combined electrical field after the second 1:N coupler can be written as
EC(Ω)=EP(Ω)+E1(Ω)+E2(Ω).
The transfer function of the DCF can be expressed as
T(Ω)=|T(Ω)|exp[jΦ(Ω)].
The loss of optical fiber within the RF range can be considered as constant, consequently, the DCF can be regarded as a phase filter, i.e., |T(Ω)|=1 and Φ(Ω) can be expanded based on Taylor expansion,
Φ(Ω)=Φ(Ω0)+τ0(ΩΩ0)+12DΩ(ΩΩ0)2,
where τ0 and DΩ are the group delay and GVD centered at Ω0, respectively. The spectrum of the electrical field before the PD can be written as
EPD(Ω)=EC(Ω)T(Ω).
The PD converts the optical signal to an electrical signal which can be written as
I(t)=ePD(t)ePD*(t).
Where ℜ is responsivity of the PD. Applying the Fourier transform to both sides of Eq. (12), the spectrum of the electrical signal can be written as
I(ω)=12πEPD(ω)EPD*(ω)=2πEC(Ω)EC*(Ωω)T(Ω)T*(Ωω)dΩ,
where denotes convolution and ω is the angular frequency of the RF signal. Considering the frequency-domain representation of the input RF signal is πVe[δ(ω-ωe) + δ(ω-ωe)], then the MPF’s transfer function can be written as
HRF(ω)=I(ω)/{πVe[δ(ωωe)+δ(ω+ωe)]}=H0(ω)+H1(ω)+H2(ω)
where
H0(ω)=2kP2J0(γ)J1(γ)πVeexp(jωτ0)sin(DΩω2/2)Hb(ω),
H1(ω)=2k1kPJ1(γ)πVeexp(ωτ0)N(Ω)exp[jDΩ(ΩΩ0)ω]sin(Ωt1DΩω2/2)dΩ=k1kPJ1(γ)πVeexpj(π/2ωτ0DΩω2/2+t1Ω0)Hb(ωt1DΩ)+k1kPJ1(γ)πVeexpj(π/2ωτ0+DΩω2/2t1Ω0)Hb(ω+t1DΩ)
H2(ω)=2k1kPJ1(γ)πVeexp(ωτ0)N(Ω)exp[jDΩ(ΩΩ0)ω]sin(Ωt2DΩω2/2)dΩ=k2kPJ1(γ)πVeexpj(π/2ωτ0DΩω2/2+t2Ω0)Hb(ωt2DΩ)+k2kPJ1(γ)πVeexpj(π/2ωτ0+DΩω2/2t2Ω0)Hb(ω+t2DΩ),
where Hb(ω)=N(Ω)exp[jωDΩ(ΩΩ0)]dΩ stands for a baseband response. In writing Eq. (14), the DC and small second harmonic components are ignored. The beating of carrrers without delay and delayed by t1 and t2 only contributes to the intensity noise. The detailed derivation of Eq. (14) is given in Appendix. In Eq. (14), H1(ω) is the baseband response eliminated by the CSE. Therefore the MPF only have two passbands H1(ω) and H2(ω whose amplitude responses can be seen as frequency-shifted versions of the baseband response centered at t1/DΩ and t1/DΩ. By adjusting the time delay introduced by the corresponding ODL, the center frequency of each passband can be independently tuned.

The MPF can be treated as a special case of the MPF based on optical delay line structure. Previously, the BOS is sliced by AWG or FBG to generate taps of the MPF [19, 20]. Without any explicit slicing procedure, the BOS itself can be ideally sliced to an array of lightwaves which have infinitesimal wavelength intervals. Furthermore, their phases are uncorrelated according to the statistic properties of BOS as shown in Eq. (3). A response can only come from the beating of sidebands generated by phase modulation and the carriers with or without delay. The optical signal before and after the DCF is illustrated in Fig. 2. Every ideally sliced lightwave forms a tap in the MPF. For one tap, the response originated from the beating of sidebands and the carrier transmitted through a dispersive medium [21] can be given by

Hn(ω)N(Ωn)ΔΩsin(ΩntDΩω2/2)exp[jωτd(Ωn)]
where Ωn is the angular frequency of a sliced lightwave from the BOS, N(Ωn)ΔΩ is the power of the sliced lightwave, is the time delay of the carrier relative to sidebands, τd is the group delay of the lightwave through the DCF.

 figure: Fig. 2

Fig. 2 Illustration of the optical signal before and after the DCF. Blue lines: the optical signal after phase modulating. Red lines: the optical signal delayed by ODL1. Yellow lines: the optical signal delayed by ODL2.

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The BOS comprise many sliced lightwaves, as a result, the overall transfer function of the MPF can be given by [2]

HRF(ω)n=[N(Ωn)ΔΩsin(ΩntDΩω2/2)]exp[jωτd(Ωn)]=N(Ω)sin(ΩtDΩω2/2)exp[jωτd(Ω)]dΩ=exp(jωτ0)N(Ω)sin(ΩtDΩω2/2)exp[jωDΩ(ΩΩ0)]dΩ.

When ΔΩ approaches to dΩ, the sum converges to integral and the time delay difference between adjacent taps approaches to infinitesimal. Inherently, MPFs with delay structure has periodic passbands whose FSR is inverse proportional to the time delay difference. As a result, if the delay difference approaches infinitesimal, a single passband will be formed. Equation (16) can be also written as

HRF(ω)j2n=[N(Ωn)ΔΩnexpj(ΩntDΩω2/2)]exp[jωτd(Ωn)]+j2n=[N(Ωn)ΔΩnexpj(Ωnt+DΩω2/2)]exp[jωτd(Ωn)]

When an ODL is tuned, a linear phase shift Ωn, i.e. a complex coefficient, is introduced to all taps of the MPF. According to [1], the complex coefficient can shift the center frequency of the passband while maintaining the shape of the frequency response unchanged. As each tap should be tuned independently to introduce a linear phase shift, previously, most of the MPF with complex coefficient are implemented with 2 taps [22, 23] resulting in a low Q factor. Intuitively, the number of taps is proportional to the bandwidth of BOS. As a consequence, we can employ a BOS with broader bandwidth to obtain a high Q factor as well as a low cost solution. Furthermore, by bringing in more ODLs, the center frequency of the single passband can be shifted to different positions leading to a multiple passbands operation.

3. Experimental results

An experiment based on the setup shown in Fig. 1 is performed. Two 1:3 couplers are employed in the experiment. A PM (EOSPACE PM-DV5-40-PFU-PUF-LV) with a bandwidth of 40 GHz and a half-wave voltage of 4 V is incorporated in one branch between the two couplers. The time delay of the ODL (General Photonics VariDelay) can be tuned from 0 to 600 ps. The GVD of the DCF module is −989 ps/nm. The bandwidth of the PD (u2t XPDU2120R) is 50 GHz. The ASE spectrum after the OF is shown in Fig. 3. The 3-dB bandwidth of the spectrum is measured to be 3.6 nm centered at 1551.25 nm.

 figure: Fig. 3

Fig. 3 The optical spectrum of the BOS after OF.

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We approximate the shape of the BOS as rectangular which can be expressed as

N(Ω)={N|ΩΩ0|<ΔΩ/20other,
where Ω0 and ΔΩ are the center frequency and bandwidth of the BOS, respectively. Then Hb(ω) has an analytic expression which can be written as
Hb(ω)=NΔΩsinc(ωDΩΔΩ/2).
When ODL1 and ODL2 are tuned to 154.2 ps and 225 ps, the dual-passbands centered at 8 GHz and 14 GHz are shown in Fig. 4.

 figure: Fig. 4

Fig. 4 The frequency response with two passbands.

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The zoom-in view of the two passbands which are normalized are shown in Fig. 5. The simulation responses are also plotted in the same figure. As can be seen from Fig. 5, the experimental results agree well with the simulation ones. Obviously, the shape of the passband can be reconfigured by tailoring the optical spectrum. If the OF has a Gaussian shape, a better out-of-band suppression ratio can be achieved [8]. Based on Eq. (18), the bandwidth of the passband can be decreased by increasing the bandwidth of the OF or the length of the DCF. It is worth noting that the bandwidth of the passband cannot be decreased infinitely as the dispersion slope or the third-order dispersion of the DCF cannot be neglected when the bandwidth of the OF is wide. The dispersion slope of the DCF can be either compensated by a Waveshaper [24] or a chirped FBG [25] or avoided by replacing the DCF with a linear chirped FBG which has a low dispersion slope [8].

 figure: Fig. 5

Fig. 5 The measured (black line) and simulated (red line) bandpass responses centered at 8 GHz and 14 GHz, respectively. The magnitude is normalized.

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To illustrate the independent tunability of the dual-passband, we firstly fix the center frequency of the 1st passband at 1 GHz and the center frequency of the 2nd passband is tuned from 4 GHz to 30 GHz by tuning ODL2 from 145.5 ps to 352.2 ps with a tuning step of 15.9 ps. The results are shown in Fig. 6(a). Then we fix the center frequency of the 2nd passband at 1 GHz and the center frequency of the 1st passband is tuned from 4 GHz to 30 GHz by tuning ODL1 from 122.4 ps to 329.1 ps with a tuning step of 15.9 ps, and the results are shown in Fig. 6(b). The 3-dB bandwidths of the two passbands are about 250 MHz from 4 GHz to 30 GHz, and the magnitude of the passbands decreases by 4.467 dB and 4.71 dB for the 1st and 2nd passband from 4 GHz to 30 GHz. The 3-dB tuning ranges are 22 GHz and 20 GHz for the 1st and 2nd passbands, respectively. The ODLs have very good tunability, however the magnitude of the passband decreases with the increasing of the center frequency due to the limited bandwidth of the PM and the PD as well as the dispersion slope of the DCF [7]. If the PM and the PD with higher bandwidth are employed and the dispersion slope of the DCF is resolved by the methods mentioned above [8, 24, 25], the tuning range can be further extended.

 figure: Fig. 6

Fig. 6 (a) The 1st passband is fixed while the 2nd passband is tuned. (b) The 2nd passband is fixed while the 1st passband is tuned.

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The stability of the MPF is also studied. The center frequencies and the magnitude of the two passbands are measured every 5 minutes for 1.5 hours. The results are shown in Fig. 7. The frequency and magnitude variations of the 1st passband centered at 8 GHz are within 31 MHz and 0.4 dB, respectively. The frequency and magnitude variations of the 2nd passband centered at 14 GHz are within 37 MHz and 0.34 dB, respectively. We believe the stability of the MPF can be further improved through mechanical isolation of the interferometry structure between two 1:N couplers [26].

 figure: Fig. 7

Fig. 7 The stability of the center frequency and magnitude of (a) 1st passband and (b) 2nd passband in 1.5 h with 5 min interval.

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Finally, the SFDR of the MPF is evaluated. The SFDR of the 1st passband centered at 1 GHz is shown in Fig. 8(a) while that of 2nd passaband at 4 GHz is shown in Fig. 8(b). The measured SFDRs are 73.5 dB·Hz2/3 and 73 dB·Hz2/3, respectively. The noise floor is measured to be −125 dBm/Hz which is relatively high due to the excess intensity noise produced by the beating of Fourier components within the BOS spectrum. The SFDR can be enhanced by employing a balanced detection scheme as the signal amplitude is double and the noises are reduced at the output of the balanced photodetector [27].

 figure: Fig. 8

Fig. 8 The measured fundamental power and the third order intermodulation power for the (a) 1st passband and (b) 2nd passband of the MPF.

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4. Discussions and conclusions

We conceive an alternative configuration of an MPF with multiple passbands based on a BOS which is illustrated in Fig. 9. Different from the MPF shown in Fig. 1, here the PM is removed from one branch between the couplers. The other branches are kept unchanged with the previous configuration. The combined optical signal after the second 1:N coupler is phase modulated and sent to a DCF, after which a PD is employed to convert the optical signal to an electrical signal.

 figure: Fig. 9

Fig. 9 An alternative configuration of MPF with multiple passbands based on a BOS.

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The spectrum of electrical field after the second coupler can be expressed as

EC(Ω)=E(Ω)+exp(jt1Ω)E(Ω)+exp(jt2Ω)E(Ω).
The spectrum of electrical field after the PM can be expressed as
EP(Ω)=J0(γ)EC(Ω)+jJ1(γ)EC(Ωωe)+jJ1(γ)EC(Ω+ωe)=J0(γ)E(Ω)+jJ1(γ)E(Ωωe)+jJ1(γ)E(Ω+ωe)+J0(γ)exp(jt1Ω)E(Ω)+jJ1(γ)exp[jt1(Ωωe)]E(Ωωe)+jJ1(γ)exp[jt1(Ω+ωe)]E(Ω+ωe)+J0(γ)exp(jt2Ω)E(Ω)+jJ1(γ)exp[jt2(Ωωe)]E(Ωωe)+jJ1(γ)exp[jt2(Ω+ωe)]E(Ω+ωe).
Following the similar derivation process of Eq. (14), the MPF’s transfer function can be given by
HRF(ω)=H0(ω)+H1(ω)+H2(ω)+H3(ω),
where

H0(ω)=3jJ0(γ)J1(γ)πVeN(Ω)[T(Ω+ω)T*(Ω)T(Ω)T*(Ωω)]dΩ=6J0(γ)J1(γ)πVeexp(jτdω)sin(DΩω2/2)Hb(ω),
H1+(ω)=jJ0(γ)J1(γ)πVeexp(jτ1Ω)N(Ω)[T(Ω+ω)T*(Ω)T(Ω)T*(Ωω)]dΩ=2J0(γ)J1(γ)πVeexpj(t1Ω0τ0ω)sin(DΩω2/2)Hb(ωt1DΩ),
H2+(ω)=jJ0(γ)J1(γ)πVeexp(jτ2Ω)N(Ω)[T(Ω+ω)T*(Ω)T(Ω)T*(Ωω)]dΩ=2J0(γ)J1(γ)πVeexpj(t2Ω0τ0ω)sin(DΩω2/2)Hb(ωt2DΩ),
H3+(ω)=jJ0(γ)J1(γ)πVeexp[j(t2t1)Ω]N(Ω)[T(Ω+ω)T*(Ω)T(Ω)T*(Ωω)]dΩ=2J0(γ)J1(γ)πVeexpj[(t2t1)Ω0τ0ω]sin(DΩω2/2)Hb(ωt2t1DΩ).

The response centered at the positive frequency range is given above while the mathematical responses centered at symmetric negative frequency range are ignored for simplicity. As can be seen from Eq. (21), except the passbands centered at t1/DΩ and t2/DΩ, there appears an extra passband centered at (t2- t1)/DΩ. The BL without delay and delayed by t1 and t2 are all split into carrier and sidebands in this configuration as is shown in Fig. 10. The beating of the BL which is delayed by t1 and t2 and split into carrier and sidebands generates the third passband centered at (t2- t1)/DΩ. As a result, if more delayed optical signals are added in the MPF, any two delayed components will bring an extra passband. What can be also seen from Eq. (21) is that the magnitude of passbands are shaped by CSE. If the center frequency of the passband is located in the notch of CSE curve, the magnitude of the passband will be severely attenuated.

 figure: Fig. 10

Fig. 10 Illustration of the optical signal before and after the DCF. Blue lines: the optical signal after phase modulating. Red lines: the optical signal delayed by ODL1. Yellow lines: the optical signal delayed by ODL2.

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An experiment is carried out to confirm the theoretical analysis. The center frequency of the dual-passband is tuned to 7.95 GHz and 12 GHz, respectively. As is shown in Fig. 11(a), an extra passband centered at 4.05 GHz appears and the magnitude of passbands is shaped by the CSE. Then we tune the 2nd passband closer to the 1st passband. As the center frequency of the 2nd passband locates in the notch of the CSE curve, the magnitude of the 2nd passband is severely attenuated as is shown in Fig. 11(b). The extra passband shifts to lower frequency correspondingly and its magnitude decreases along with the CSE curve. In a word, this MPF based on a BOS does not behave well for multiple passbands operation.

 figure: Fig. 11

Fig. 11 Measured frequency response (blue line) and the simulation CSE curve (red line). The magnitude is normalized.

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In conclusion, we proposed a novel MPF with multiple independently tunable passbands which can be used in multiband/multifunctional microwave systems. A full theoretical analysis was presented and a proof-of-concept experiment to verify the MPF with dual-passband was performed. The experimental results agree well with the theory. Each of the passband can be tuned to 30 GHz. The number of passbands can be further extended. An alternative MPF configuration with multiple passbands based on a BOS was also theoretically analyzed and experimentally demonstrated. However, the performance of this MPF suffers from the extra passbands and the CSE, which is not as good as the MPF proposed before.

Appendix

In this Appendix, we develop the expression of Eq. (14). Substitute Eq. (8) to Eq. (13), the electrical signal can be expressed as

I(ω)=2πEC(Ω)EC*(Ωω)T(Ω)T*(Ωω)dΩ=2π[kPJ0(γ)E(Ω)+jkPJ1(γ)E(Ωωe)+jkPJ1(γ)E(Ω+ωe)+E1(Ω)+E2(Ω)][kPJ0(γ)E*(Ωω)jkPJ1(γ)E*(Ωωωe)jkPJ1(γ)E*(Ωω+ωe)+E1*(Ωω)+E2*(Ωω)]T(Ω)T*(Ωω)dΩ
The spectrum of the baseband electrical signal can be given by
I0(ω)=2πjkp2J0(γ)J1(γ)E(Ω)E*(Ωωωe)jkp2J0(γ)J1(γ)E(Ω)E*(Ωω+ω)e+jkp2J0(γ)J1(γ)E(Ωωe)E*(Ωω)+jkp2J0(γ)J1(γ)E(Ω+ωe)E*(Ωω)T(Ω)T*(Ωω)dΩ=2πjkp2J0(γ)J1(γ)[N(Ω)N(Ωωωe)2πδ(ω+ωe)N(Ω)N(Ωω+ωe)2πδ(ωωe)+N(Ωωe)N(Ωω)2πδ(ωωe)+N(Ω+ωe)N(Ωω)2πδ(ω+ωe)]T(Ω)T*(Ωω)dΩ=jkP2J0(γ)J1(γ)N(Ω)[H(Ω+ω)H*(Ω)H(Ω)H*(Ωω)]dΩ[δ(ω+ωe)+δ(ωωe)]
Recall the spectrum of the input signal is πVe[δ(ω-ωe) + δ(ω-ωe)], thus the baseband response can be written as
H0(ω)=I0(ω)πVe[δ(ωωe)+δ(ω+ωe)]=2kP2J0(γ)J1(γ)πVeexp(jωτ0)sin(DΩω2/2)Hb(ω)
Then we consider the bandpass response center at the t1/DΩ which comes from the beating of sidenbands and the carrier delay by t1. The corresponding electrical signal can be expressed as
I1(ω)=2πE1*(Ωω)jkpJ1(γ)E(Ωωe)+E1*(Ωω)jkpJ1(γ)E(Ω+ωe)+E1(Ω)(jkpJ(γ))E*(Ωωωe)+E1(Ω)(jkpJ(γ))E*(Ωω+ωe)T(Ω)T*(Ωω)dΩ=2πjk1kpJ1(γ){exp[j(Ωω)τ1]N(Ωω)N(Ωωe)2πδ(ωωe)T(Ω)T*(Ωω)dΩ+exp[j(Ωω)τ1]N(Ωω)N(Ω+ωe)2πδ(ω+ωe)T(Ω)T*(Ωω)dΩexp(jΩτ1)N(Ω)N(Ωωωe)2πδ(ω+ωe)T(Ω)T*(Ωω)dΩexp(jΩτ1)N(Ω)N(Ωω+ωe)2πδ(ωωe)T(Ω)T*(Ωω)dΩ}=jk1kpJ1(γ){[δ(ωωe)+δ(ω+ωe)]exp[j(Ωω)τ1]N(Ωω)T(Ω)T*(Ωω)dΩ[δ(ωωe)+δ(ω+ωe)]exp(jΩτ1)N(Ω)T(Ω)T*(Ωω)dΩ}=jk1kpJ1(γ){N(Ω)[exp(jΩτ1)T(Ω+ω)T*(Ω)exp(jΩτ1)T(Ω)T*(Ωω)]dΩ}[δ(ωωe)+δ(ω+ωe)]
The bandpass response can be written as
H1(ω)=I1(ω)πVe[δ(ωωe)+δ(ω+ωe)]=2k1kPJ1(γ)πVeexp(ωτ0)N(Ω)exp[jDΩ(ΩΩ0)ω]sin(Ωt1DΩω2/2)dΩ
The passband centered at t2/DΩ has a similar derivation process. Finally, we derive the expression of the averaged electrical signal generated by the beating of carriers delayed by t1 and t2 which can be given by
Iint(ω)=2πE1(Ω)E2*(Ωω)+E1*(Ωω)E2(Ω)T(Ω)T*(Ωω)dΩ=2π[k1k2exp(jΩτ1)exp[j(Ωω)τ2]E(Ω)E*(Ωω)+k1k2exp[j(Ωω)τ1]exp(jΩτ2)E(Ω)E*(Ωω)]T(Ω)T*(Ωω)dΩ=2πk1k2{exp[jΩτ1+j(Ωω)τ2]+exp[j(Ωω)τ1jΩτ2]}N(Ω)N(Ωω)2πδ(ω)T(Ω)T*(Ωω)dΩ=k1k22cos[Ω(τ2τ1)]N(Ω)T(Ω)T*(Ω)dΩδ(ω)
As can be seen from (27), the averaged beating signal of the carriers delayed by t1 and t2 only contribute to the DC component, therefore more passbands can be formed by introducing more time-shifted carriers.

Acknowledgment

This work was supported in part by the Technology Support Program of Jiangsu Province under grant BE2012157; the National Natural Science Foundation of China (NSFC) under grants 61475193, 61032005, 61177065; and the Jiangsu Province Natural Science Foundationunder grant BK2012058, BK20140069. The authors would like to thank Shifeng Liu for lending the optical delay lines.

References and links

1. J. Yao, “Microwave photonics,” J. Lightwave Technol. 54(2), 832–846 (2006).

2. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006). [CrossRef]  

3. W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012). [CrossRef]  

4. B. Vidal, M. A. Piqueras, and J. Martí, “Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering,” Opt. Lett. 32(1), 23–25 (2007). [CrossRef]   [PubMed]  

5. J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010). [CrossRef]  

6. W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012). [CrossRef]  

7. J. Mora, B. Ortega, A. Diez, J. L. Cruz, M. V. Andres, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on Mach-Zehnder interferometer,” J. Lightwave Technol. 24(7), 2500–2509 (2006). [CrossRef]  

8. L. Li, X. Yi, T. X. H. Huang, and R. A. Minasian, “Shifted dispersion-induced radio-frequency fading in microwave photonic filters using a dual-input Mach-Zehnder electro-optic modulator,” Opt. Lett. 38(7), 1164–1166 (2013). [CrossRef]   [PubMed]  

9. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Widely tunable single-bandpass microwave photonic filter employing a non-sliced broadband optical source,” Opt. Express 19(19), 18423–18429 (2011). [CrossRef]   [PubMed]  

10. R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015). [CrossRef]  

11. G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013). [CrossRef]  

12. L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014). [CrossRef]  

13. J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010). [CrossRef]  

14. Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013). [CrossRef]  

15. H. Chen, Z. Xu, H. Fu, S. Zhang, C. Wu, H. Wu, H. Xu, and Z. Cai, “Switchable and tunable microwave frequency multiplication based on a dual-passband microwave photonic filter,” Opt. Express 23(8), 9835–9843 (2015). [CrossRef]   [PubMed]  

16. L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014). [CrossRef]  

17. X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

18. G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37(8), 1008–1014 (2001). [CrossRef]  

19. D. Pastor, B. Ortega, J. Capmany, S. Sales, A. Martinez, and P. Muñoz, “Optical microwave filter based on spectral slicing by use of arrayed waveguide gratings,” Opt. Lett. 28(19), 1802–1804 (2003). [CrossRef]   [PubMed]  

20. M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005). [CrossRef]  

21. F. Zeng and J. Yao, “All-optical bandpass microwave filter based on an electro-optic phase modulator,” Opt. Express 12(16), 3814–3819 (2004). [CrossRef]   [PubMed]  

22. Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013). [CrossRef]  

23. A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006). [CrossRef]  

24. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Analysis and compensation of third-order dispersion induced RF distortions in highly reconfigurable microwave photonic filters,” J. Lightwave Technol. 31(13), 2263–2270 (2013). [CrossRef]  

25. G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015). [CrossRef]  

26. D. W. Grund Jr, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Improved configuration and reduction of phase noise in a narrow linewidth ultrawideband optical RF source,” Opt. Lett. 39(16), 4667–4670 (2014). [CrossRef]   [PubMed]  

27. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Highly reconfigurable microwave photonic single-bandpass filter with complex continuous-time impulse responses,” Opt. Express 20(24), 26929–26934 (2012). [CrossRef]   [PubMed]  

References

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  1. J. Yao, “Microwave photonics,” J. Lightwave Technol. 54(2), 832–846 (2006).
  2. J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006).
    [Crossref]
  3. W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
    [Crossref]
  4. B. Vidal, M. A. Piqueras, and J. Martí, “Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering,” Opt. Lett. 32(1), 23–25 (2007).
    [Crossref] [PubMed]
  5. J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
    [Crossref]
  6. W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
    [Crossref]
  7. J. Mora, B. Ortega, A. Diez, J. L. Cruz, M. V. Andres, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on Mach-Zehnder interferometer,” J. Lightwave Technol. 24(7), 2500–2509 (2006).
    [Crossref]
  8. L. Li, X. Yi, T. X. H. Huang, and R. A. Minasian, “Shifted dispersion-induced radio-frequency fading in microwave photonic filters using a dual-input Mach-Zehnder electro-optic modulator,” Opt. Lett. 38(7), 1164–1166 (2013).
    [Crossref] [PubMed]
  9. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Widely tunable single-bandpass microwave photonic filter employing a non-sliced broadband optical source,” Opt. Express 19(19), 18423–18429 (2011).
    [Crossref] [PubMed]
  10. R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015).
    [Crossref]
  11. G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
    [Crossref]
  12. L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
    [Crossref]
  13. J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
    [Crossref]
  14. Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
    [Crossref]
  15. H. Chen, Z. Xu, H. Fu, S. Zhang, C. Wu, H. Wu, H. Xu, and Z. Cai, “Switchable and tunable microwave frequency multiplication based on a dual-passband microwave photonic filter,” Opt. Express 23(8), 9835–9843 (2015).
    [Crossref] [PubMed]
  16. L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
    [Crossref]
  17. X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.
  18. G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37(8), 1008–1014 (2001).
    [Crossref]
  19. D. Pastor, B. Ortega, J. Capmany, S. Sales, A. Martinez, and P. Muñoz, “Optical microwave filter based on spectral slicing by use of arrayed waveguide gratings,” Opt. Lett. 28(19), 1802–1804 (2003).
    [Crossref] [PubMed]
  20. M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
    [Crossref]
  21. F. Zeng and J. Yao, “All-optical bandpass microwave filter based on an electro-optic phase modulator,” Opt. Express 12(16), 3814–3819 (2004).
    [Crossref] [PubMed]
  22. Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
    [Crossref]
  23. A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
    [Crossref]
  24. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Analysis and compensation of third-order dispersion induced RF distortions in highly reconfigurable microwave photonic filters,” J. Lightwave Technol. 31(13), 2263–2270 (2013).
    [Crossref]
  25. G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
    [Crossref]
  26. D. W. Grund, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Improved configuration and reduction of phase noise in a narrow linewidth ultrawideband optical RF source,” Opt. Lett. 39(16), 4667–4670 (2014).
    [Crossref] [PubMed]
  27. X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Highly reconfigurable microwave photonic single-bandpass filter with complex continuous-time impulse responses,” Opt. Express 20(24), 26929–26934 (2012).
    [Crossref] [PubMed]

2015 (3)

R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015).
[Crossref]

H. Chen, Z. Xu, H. Fu, S. Zhang, C. Wu, H. Wu, H. Xu, and Z. Cai, “Switchable and tunable microwave frequency multiplication based on a dual-passband microwave photonic filter,” Opt. Express 23(8), 9835–9843 (2015).
[Crossref] [PubMed]

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
[Crossref]

2014 (3)

D. W. Grund, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Improved configuration and reduction of phase noise in a narrow linewidth ultrawideband optical RF source,” Opt. Lett. 39(16), 4667–4670 (2014).
[Crossref] [PubMed]

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

2013 (5)

G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
[Crossref]

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

L. Li, X. Yi, T. X. H. Huang, and R. A. Minasian, “Shifted dispersion-induced radio-frequency fading in microwave photonic filters using a dual-input Mach-Zehnder electro-optic modulator,” Opt. Lett. 38(7), 1164–1166 (2013).
[Crossref] [PubMed]

X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Analysis and compensation of third-order dispersion induced RF distortions in highly reconfigurable microwave photonic filters,” J. Lightwave Technol. 31(13), 2263–2270 (2013).
[Crossref]

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

2012 (3)

X. Xue, X. Zheng, H. Zhang, and B. Zhou, “Highly reconfigurable microwave photonic single-bandpass filter with complex continuous-time impulse responses,” Opt. Express 20(24), 26929–26934 (2012).
[Crossref] [PubMed]

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
[Crossref]

2011 (1)

2010 (2)

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
[Crossref]

2007 (1)

2006 (4)

J. Yao, “Microwave photonics,” J. Lightwave Technol. 54(2), 832–846 (2006).

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24(1), 201–229 (2006).
[Crossref]

J. Mora, B. Ortega, A. Diez, J. L. Cruz, M. V. Andres, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on Mach-Zehnder interferometer,” J. Lightwave Technol. 24(7), 2500–2509 (2006).
[Crossref]

A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
[Crossref]

2005 (1)

M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
[Crossref]

2004 (1)

2003 (1)

2001 (1)

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37(8), 1008–1014 (2001).
[Crossref]

Andres, M. V.

Bai, G.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Cai, Z.

Capmany, J.

Chang, D.-C.

J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
[Crossref]

Chaudhary, G.

G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
[Crossref]

Chen, H.

Chen, X.

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

Cruz, J. L.

Diez, A.

Duan, G.-H.

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37(8), 1008–1014 (2001).
[Crossref]

Fernandez-Pousa, C. R.

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
[Crossref]

Fonjallaz, P.-Y.

M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
[Crossref]

Fu, H.

Galan, J. V.

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Gao, L.

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

Georgiev, E.

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37(8), 1008–1014 (2001).
[Crossref]

Gomez-Garcia, R.

R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015).
[Crossref]

Grund, D. W.

Gunnarsson, O.

M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
[Crossref]

Guyette, A. C.

R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015).
[Crossref]

Han, X.

X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

Hu, B.-J.

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

Huang, T. X. H.

Jeong, Y.

G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
[Crossref]

Jiang, Y.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Li, H.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Li, L.

Li, M.

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

Li, W.

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
[Crossref]

Lim, J.

G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
[Crossref]

Liu, J.-C.

J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
[Crossref]

Liu, W.

X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

Loayssa, A.

A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
[Crossref]

Maestre, H.

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
[Crossref]

Marti, J.

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
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A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
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Murakowski, J.

Ortega, B.

Palaci, J.

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
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Pan, S.

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
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Pastor, D.

Peraita, G.

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
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Piqueras, M. A.

Popov, M.

M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
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Prather, D. W.

Sagues, M.

A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
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Schneider, G. J.

Shi, S.

Shum, P. P.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Torregrosa, A. J.

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
[Crossref]

Vidal, B.

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
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J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Wang, J.-W.

J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
[Crossref]

Wang, S.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Wu, C.

Wu, H.

Xu, E.

X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

Xu, H.

Xu, J.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Xu, Z.

Xue, Q.

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

Xue, X.

Yao, J.

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
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X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

Yao, J. P.

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

Yi, X.

Zeng, B.-H.

J.-C. Liu, J.-W. Wang, B.-H. Zeng, and D.-C. Chang, “CPW-fed dual-mode double-square-ring resonators for quad-band filters,” IEEE Microw. Wirel. Compon. Lett. 20(3), 142–144 (2010).
[Crossref]

Zeng, F.

Zhang, H.

Zhang, J.

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

Zhang, S.

Zhang, X. Y.

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

Zhang, Y.

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

Zheng, X.

Zhou, B.

Zhou, J.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Zhou, Z.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

Zu, P.

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
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IEEE Microw. Wirel. Compon. Lett. (1)

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[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Popov, P.-Y. Fonjallaz, and O. Gunnarsson, “Compact microwave photonic transversal filter with 40-dB sidelobe suppression,” IEEE Photon. Technol. Lett. 17(3), 663–665 (2005).
[Crossref]

IEEE Photonics J. (1)

Y. Jiang, P. P. Shum, P. Zu, J. Zhou, G. Bai, J. Xu, Z. Zhou, H. Li, and S. Wang, “A selectable multiband bandpass microwave photonic filter,” IEEE Photonics J. 5(3), 5500509 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (4)

J. Palaci, G. E. Villanueva, J. V. Galan, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photonics Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Y. Zhang and S. Pan, “Complex coefficient microwave photonic filter using a polarization-modulator-based phase shifter,” IEEE Photonics Technol. Lett. 25(2), 187–189 (2013).
[Crossref]

A. Loayssa, J. Capmany, M. Sagues, and J. Mora, “Demonstration of incoherent microwave photonic filters with all-optical complex coefficients,” IEEE Photonics Technol. Lett. 18(16), 1744–1746 (2006).
[Crossref]

G. Peraita, A. J. Torregrosa, H. Maestre, and C. R. Fernandez-Pousa, “Broadband linearization of dispersive delay line using a chirped fiber bragg grating,” IEEE Photonics Technol. Lett. 27(10), 1044–1135 (2015).
[Crossref]

IEEE Trans. Microw. Theory Tech. (6)

W. Li, M. Li, and J. P. Yao, “A narrow-passband and frequency-tunable micro-wave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(5), 1287–1296 (2012).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
[Crossref]

R. Gomez-Garcia and A. C. Guyette, “Reconfigurable multi-band microwave filters,” IEEE Trans. Microw. Theory Tech. 63(4), 1294–1307 (2015).
[Crossref]

G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Tech. 61(1), 107–116 (2013).
[Crossref]

L. Gao, X. Y. Zhang, B.-J. Hu, and Q. Xue, “Novel multi-stub loaded resonators and their applications to various bandpass filters,” IEEE Trans. Microw. Theory Tech. 62(5), 1162–1172 (2014).
[Crossref]

L. Gao, J. Zhang, X. Chen, and J. Yao, “Microwave photonic filter with two independently tunable passbands using a phase modulator and an equivalent phase-shifted fiber bragg grating,” IEEE Trans. Microw. Theory Tech. 62(2), 380–387 (2014).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (4)

Opt. Lett. (4)

Other (1)

X. Han, E. Xu, W. Liu, and J. Yao, “Tunable dual-passband microwave photonic filter using orthogonal polarization modulation,” IEEE Photon. Technol. Lett.to be published.

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of the proposed MPF with multiple independently tunable passbands. Broadband optical source: BOS, optical filter: OF, optical delay line: ODL, variable optical attenuator: VOA, phase modulator: PM, polarization controller: PC, dispersion compensation fiber: DCF, erbium-doped fiber amplifier: EDFA, photodetector: PD, vector network analyzer: VNA.
Fig. 2
Fig. 2 Illustration of the optical signal before and after the DCF. Blue lines: the optical signal after phase modulating. Red lines: the optical signal delayed by ODL1. Yellow lines: the optical signal delayed by ODL2.
Fig. 3
Fig. 3 The optical spectrum of the BOS after OF.
Fig. 4
Fig. 4 The frequency response with two passbands.
Fig. 5
Fig. 5 The measured (black line) and simulated (red line) bandpass responses centered at 8 GHz and 14 GHz, respectively. The magnitude is normalized.
Fig. 6
Fig. 6 (a) The 1st passband is fixed while the 2nd passband is tuned. (b) The 2nd passband is fixed while the 1st passband is tuned.
Fig. 7
Fig. 7 The stability of the center frequency and magnitude of (a) 1st passband and (b) 2nd passband in 1.5 h with 5 min interval.
Fig. 8
Fig. 8 The measured fundamental power and the third order intermodulation power for the (a) 1st passband and (b) 2nd passband of the MPF.
Fig. 9
Fig. 9 An alternative configuration of MPF with multiple passbands based on a BOS.
Fig. 10
Fig. 10 Illustration of the optical signal before and after the DCF. Blue lines: the optical signal after phase modulating. Red lines: the optical signal delayed by ODL1. Yellow lines: the optical signal delayed by ODL2.
Fig. 11
Fig. 11 Measured frequency response (blue line) and the simulation CSE curve (red line). The magnitude is normalized.

Equations (35)

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e(t)= 1 2π E(Ω)exp(jΩt)dΩ ,
E(Ω)= N(Ω) expjθ(Ω),
<expjθ(Ω)>=0 <expj[θ(Ω)θ( Ω )]>=2πδ(Ω Ω ),
e (t) P = k P e(t)exp[ jγcos( ω e t) ]= k P e(t) n= j n J n (γ)exp(jn ω e t) k P e(t)[ J 0 (γ)+j J 1 (γ)exp(j ω e t)+j J 1 (γ)exp(j ω e t)],
E P (Ω)= k P J 0 (γ)E(Ω)+j k P J 1 (γ)E(Ω ω e )+j k P J 1 (γ)E(Ω+ ω e ).
E 1 (Ω)= k 1 exp(jΩ t 1 )E(Ω).
E 2 (Ω)= k 2 exp(jΩ t 2 )E(Ω).
E C (Ω)= E P (Ω)+ E 1 (Ω)+ E 2 (Ω).
T(Ω)=| T(Ω) |exp[jΦ(Ω)].
Φ(Ω)=Φ( Ω 0 )+ τ 0 (Ω Ω 0 )+ 1 2 D Ω (Ω Ω 0 ) 2 ,
E PD (Ω)= E C (Ω)T(Ω).
I(t)= e PD (t) e PD * (t) .
I(ω)= 1 2π E PD (ω) E PD * (ω) = 2π E C (Ω) E C * (Ωω) T(Ω) T * (Ωω)dΩ ,
H RF (ω)=I(ω)/{ π V e [ δ(ω ω e )+δ(ω+ ω e ) ] }= H 0 (ω)+ H 1 (ω)+ H 2 (ω)
H 0 (ω)= 2 k P 2 J 0 (γ) J 1 (γ) π V e exp(jω τ 0 )sin( D Ω ω 2 /2) H b (ω),
H 1 (ω)= 2 k 1 k P J 1 (γ) π V e exp(ω τ 0 ) N(Ω)exp[j D Ω (Ω Ω 0 )ω]sin(Ω t 1 D Ω ω 2 /2)dΩ = k 1 k P J 1 (γ) π V e expj(π/2ω τ 0 D Ω ω 2 /2+ t 1 Ω 0 ) H b (ω t 1 D Ω ) + k 1 k P J 1 (γ) π V e expj(π/2ω τ 0 + D Ω ω 2 /2 t 1 Ω 0 ) H b (ω+ t 1 D Ω )
H 2 (ω)= 2 k 1 k P J 1 (γ) π V e exp(ω τ 0 ) N(Ω)exp[j D Ω (Ω Ω 0 )ω]sin(Ω t 2 D Ω ω 2 /2)dΩ = k 2 k P J 1 (γ) π V e expj(π/2ω τ 0 D Ω ω 2 /2+ t 2 Ω 0 ) H b (ω t 2 D Ω ) + k 2 k P J 1 (γ) π V e expj(π/2ω τ 0 + D Ω ω 2 /2 t 2 Ω 0 ) H b (ω+ t 2 D Ω ),
H n (ω)N( Ω n )ΔΩsin( Ω n t D Ω ω 2 /2)exp[ jω τ d ( Ω n ) ]
H RF (ω) n= [ N( Ω n )ΔΩsin( Ω n t D Ω ω 2 /2) ] exp[ jω τ d ( Ω n ) ] = N(Ω)sin(Ω t D Ω ω 2 /2) exp[ jω τ d (Ω) ]dΩ =exp(jω τ 0 ) N(Ω)sin(Ω t D Ω ω 2 /2) exp[ jω D Ω (Ω Ω 0 ) ]dΩ.
H RF (ω) j 2 n= [ N( Ω n )Δ Ω n expj( Ω n t D Ω ω 2 /2) ] exp[ jω τ d ( Ω n ) ] + j 2 n= [ N( Ω n )Δ Ω n expj( Ω n t + D Ω ω 2 /2) ] exp[ jω τ d ( Ω n ) ]
N(Ω)={ N| Ω Ω 0 |<ΔΩ/2 0other ,
H b (ω)=NΔΩsinc(ω D Ω ΔΩ/2).
E C (Ω)=E(Ω)+exp(j t 1 Ω)E(Ω)+exp(j t 2 Ω)E(Ω).
E P (Ω)= J 0 (γ) E C (Ω)+j J 1 (γ) E C (Ω ω e )+j J 1 (γ) E C (Ω+ ω e ) = J 0 (γ)E(Ω)+j J 1 (γ)E(Ω ω e )+j J 1 (γ)E(Ω+ ω e ) + J 0 (γ)exp(j t 1 Ω)E(Ω)+j J 1 (γ)exp[j t 1 (Ω ω e )]E(Ω ω e )+j J 1 (γ)exp[j t 1 (Ω+ ω e )]E(Ω+ ω e ) + J 0 (γ)exp(j t 2 Ω)E(Ω)+j J 1 (γ)exp[j t 2 (Ω ω e )]E(Ω ω e )+j J 1 (γ)exp[j t 2 (Ω+ ω e )]E(Ω+ ω e ).
H RF (ω)= H 0 (ω)+ H 1 (ω)+ H 2 (ω)+ H 3 (ω),
H 0 (ω)=3j J 0 (γ) J 1 (γ) π V e N(Ω) [ T(Ω+ω) T * (Ω)T(Ω) T * (Ωω) ]dΩ = 6 J 0 (γ) J 1 (γ) π V e exp(j τ d ω)sin( D Ω ω 2 /2) H b (ω),
H 1 + (ω)=j J 0 (γ) J 1 (γ) π V e exp(j τ 1 Ω)N(Ω)[ T(Ω+ω) T * (Ω)T(Ω) T * (Ωω) ] dΩ = 2 J 0 (γ) J 1 (γ) π V e expj( t 1 Ω 0 τ 0 ω)sin( D Ω ω 2 /2) H b (ω t 1 D Ω ),
H 2 + (ω)=j J 0 (γ) J 1 (γ) π V e exp(j τ 2 Ω)N(Ω)[ T(Ω+ω) T * (Ω)T(Ω) T * (Ωω) ] dΩ = 2 J 0 (γ) J 1 (γ) π V e expj( t 2 Ω 0 τ 0 ω)sin( D Ω ω 2 /2) H b (ω t 2 D Ω ),
H 3 + (ω)=j J 0 (γ) J 1 (γ) π V e exp[j( t 2 t 1 )Ω]N(Ω)[ T(Ω+ω) T * (Ω)T(Ω) T * (Ωω) ] dΩ = 2 J 0 (γ) J 1 (γ) π V e expj[( t 2 t 1 ) Ω 0 τ 0 ω]sin( D Ω ω 2 /2) H b (ω t 2 t 1 D Ω ).
I(ω)= 2π E C (Ω) E C * (Ωω) T(Ω) T * (Ωω)dΩ = 2π [ k P J 0 (γ)E(Ω)+j k P J 1 (γ)E(Ω ω e )+j k P J 1 (γ)E(Ω+ ω e )+ E 1 (Ω)+ E 2 (Ω) ] [ k P J 0 (γ) E * (Ωω)j k P J 1 (γ) E * (Ωω ω e )j k P J 1 (γ) E * (Ωω+ ω e )+ E 1 * (Ωω)+ E 2 * (Ωω) ] T(Ω) T * (Ωω)dΩ
I 0 (ω)= 2π j k p 2 J 0 (γ) J 1 (γ)E(Ω) E * (Ωω ω e ) j k p 2 J 0 (γ) J 1 (γ)E(Ω) E * (Ωω+ω ) e +j k p 2 J 0 (γ) J 1 (γ)E(Ω ω e ) E * (Ωω)+j k p 2 J 0 (γ) J 1 (γ)E(Ω+ ω e ) E * (Ωω) T(Ω) T * (Ωω)dΩ = 2π j k p 2 J 0 (γ) J 1 (γ) [ N(Ω) N(Ωω ω e ) 2πδ(ω+ ω e ) N(Ω) N(Ωω+ ω e ) 2πδ(ω ω e ) + N(Ω ω e ) N(Ωω) 2πδ(ω ω e )+ N(Ω+ ω e ) N(Ωω) 2πδ(ω+ ω e ) ]T(Ω) T * (Ωω)dΩ =j k P 2 J 0 (γ) J 1 (γ) N(Ω) [ H(Ω+ω) H * (Ω)H(Ω) H * (Ωω) ]dΩ[ δ(ω+ ω e )+δ(ω ω e ) ]
H 0 (ω)= I 0 (ω) π V e [ δ(ω ω e )+δ(ω+ ω e ) ] = 2 k P 2 J 0 (γ) J 1 (γ) π V e exp(jω τ 0 )sin( D Ω ω 2 /2) H b (ω)
I 1 (ω)= 2π E 1 * (Ωω)j k p J 1 (γ)E(Ω ω e )+ E 1 * (Ωω)j k p J 1 (γ)E(Ω+ ω e ) + E 1 (Ω)(j k p J(γ)) E * (Ωω ω e )+ E 1 (Ω)(j k p J(γ)) E * (Ωω+ ω e ) T(Ω) T * (Ωω)dΩ = 2π j k 1 k p J 1 (γ){ exp[j(Ωω) τ 1 ] N(Ωω) N(Ω ω e ) 2πδ(ω ω e )T(Ω) T * (Ωω)dΩ + exp[j(Ωω) τ 1 ] N(Ωω) N(Ω+ ω e ) 2πδ(ω+ ω e )T(Ω) T * (Ωω)dΩ exp(jΩ τ 1 ) N(Ω) N(Ωω ω e ) 2πδ(ω+ ω e )T(Ω) T * (Ωω)dΩ exp(jΩ τ 1 ) N(Ω) N(Ωω+ ω e ) 2πδ(ω ω e )T(Ω) T * (Ωω)dΩ } =j k 1 k p J 1 (γ){ [ δ(ω ω e )+δ(ω+ ω e ) ] exp[j(Ωω) τ 1 ]N(Ωω)T(Ω) T * (Ωω)dΩ [ δ(ω ω e )+δ(ω+ ω e ) ] exp(jΩ τ 1 )N(Ω)T(Ω) T * (Ωω)dΩ } =j k 1 k p J 1 (γ){ N(Ω)[ exp(jΩ τ 1 )T(Ω+ω) T * (Ω)exp(jΩ τ 1 )T(Ω) T * (Ωω) ]dΩ } [ δ(ω ω e )+δ(ω+ ω e ) ]
H 1 (ω)= I 1 (ω) π V e [ δ(ω ω e )+δ(ω+ ω e ) ] = 2 k 1 k P J 1 (γ) π V e exp(ω τ 0 ) N(Ω)exp[j D Ω (Ω Ω 0 )ω]sin(Ω t 1 D Ω ω 2 /2)dΩ
I int (ω)= 2π E 1 (Ω) E 2 * (Ωω)+ E 1 * (Ωω) E 2 (Ω) T(Ω) T * (Ωω)dΩ = 2π [ k 1 k 2 exp(jΩ τ 1 )exp[j(Ωω) τ 2 ] E(Ω) E * (Ωω) + k 1 k 2 exp[j(Ωω) τ 1 ]exp(jΩ τ 2 ) E(Ω) E * (Ωω) ]T(Ω) T * (Ωω)dΩ = 2π k 1 k 2 { exp[jΩ τ 1 +j(Ωω) τ 2 ]+exp[j(Ωω) τ 1 jΩ τ 2 ] } N(Ω) N(Ωω) 2π δ(ω) T(Ω) T * (Ωω)dΩ = k 1 k 2 2cos[Ω( τ 2 τ 1 )] N(Ω)T(Ω) T * (Ω)dΩδ(ω)

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