## Abstract

The impact of group delay ripples of chirped fiber gratings (CFG) on the performance of optical beamforming networks (OBFN) is investigated. The paper theoretically analyzes the quantified relations among the amplitude and period of CFG, the optical angle frequency interval at the inter-element arrays and the beampointing shift. The wavelength instability of the optical source is also investigated. This instability-induced phase jitter of RF signal has been verified experimentally. The theoretical models are proposed to analyze the performance of CFG-based OBFN systems.

©2008 Optical Society of America

## 1. Introduction

Optical beamforming networks are considered to be as one of the most important area in microwave photonics [1] and have been under intensive investigation for many applications, including wideband phased array antennas [2], broadband wireless access and millimeter-wave radio local area networks [3]. OBFNs based on true-time delay (TTD) offer many advantages over traditional electronic steering systems, such as low loss, small size, electromagnetic immunity, and especially, wide instantaneous bandwidth and squint-free array steering. So far, among various TTD configurations, dispersion based TTD beam steerer[4–9] has been considered as a promising technique to drive wideband microwave antennas. CFG as a compact, reliable and mature dispersion element, has been used in such scenario. Moreover, it can provide broadband operation and continuous spatial scanning.

In the previous approaches, the researches on OBFNs based on CFG were mostly focused on the different implementation structures [6–9]. The negative effect of group delay ripples (GDR) caused by the resonant nature and the manufacture imperfection of CFG is inevitable in these approaches. Although fabrication techniques such as apodisation may bring significant improvements in the linearity, there still remains slight ripple in both reflectivity and delay characteristics. Several papers indicate how the GDR impairs the system performances of optical communication networks [10,11] and subcarrier-multiplexed (SCM) lightwave transmission systems [12]. But for the OBFNs, the effect of GDR is scarcely investigated.

In this paper, we investigate the effects of GDR on OBFNs theoretically and experimentally. In addition, the instability of optical sources and the phase noise of RF signal are considered and experimentally validated. The relationship of the amplitude and period of CFG, the scanning step of optical angle frequency between the inter-element arrays and the beam-pointing shift has been used in analyzing the performance of CFG-based OBFN systems.

## 2. Theoretical description

A general model of OBFNs based on a single CFG is shown in Fig. 1 [10]. The combined outputs of N-tunable laser are modulated with the desired RF signal by an electrooptic modulator, and the modulated signals are fed into linear CFG so that the different time delays are obtained in accordance with the particular carrier wavelengths for different N channels, which are split by a 1:N demutipexer. So we can tune simultaneously with equally increasing or decreasing wavelength spacing of tunable lasers to obtain different time delay interval of RF signal between adjacent channels and then achieve beamforming scanning. Such a co-channel dispersion topology can reduce the influence of environments and improve the system stability. In practice, the time delay spectrum of linear CFG can be expressed by:

where *D* and *ω*
_{0} are the dispersion coefficient and the central angular frequency of CFG, respectively. *λ*
_{0} is the central wavelength. *τ _{line}*(

*ω*) is the linear time response, which is proportional to

*D*, and

*τ*(

_{GDR}*ω*) is the GDR of CFG, which can be expressed as a sum of (

*n*<∞) sine functions:

where *ω* is the optical frequency, and *A*
_{g,k}, *T*
_{g,k} and *θ _{g,k}* are the random values of amplitude, period and angular offset of the

*k*th sine function, respectively. But for a given CFG, they would be fixed. Their values can be obtained from the Fourier transform of a measured GDR. The GDR leads to phase deviation of the RF signal received at an array element, which can then be described as follows:

where *ω _{RF}* represents the angular frequency of RF signal. From Eq. (3), we notice that when

*ω*is larger than

_{RF}*T*

_{g,k}, the delay time ripple amplitude of the detected RF signal corresponding to the GDR of CFG will be decreased. In addition, the instability of optical sources and the phase noise of RF signal will also affect the phase of received RF signal. The first one will result in frequency shift of optical wavelength and the other one will result in spectral broadening of RF signal, which are both random processes. Given that the random changes of optical sources frequency is Δ

*ω*(

_{c}*t*) and the random signal spectrum changes of RF signal sources is Δ

*ω*(

_{RF}*t*), according to Eq. (3), the induced phase jitterring of detected RF signal can be described as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}~[\u2012{\tau}_{1}\left({\omega}_{0}\right)+D\frac{{\lambda}_{0}^{2}\omega}{2\pi c}-\sum _{k=1}^{n}{A}_{g,k}\mathrm{sin}\left(\frac{\omega +\Delta {\omega}_{c}\left(t\right)}{{T}_{g,k}}+{\theta}_{g,k}\right)\mathrm{cos}\left(\frac{{\omega}_{RF}}{{T}_{g,k}}\right)]\u2022\Delta {\omega}_{c}\left(t\right)$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left[D\frac{{\lambda}_{0}^{2}{\omega}_{RF}}{2\pi c}-\sum _{k=1}^{n}{A}_{g,k}\frac{\mathrm{sin}\left(\frac{{\omega}_{RF}}{{T}_{g,k}}\right)}{\frac{{\omega}_{RF}}{{T}_{g,k}}}\mathrm{sin}\left(\frac{\omega +\frac{1}{2}\Delta {\omega}_{c}\left(t\right)}{{T}_{g,k}}+{\theta}_{g,k}\right)\right]\u2022\Delta {\omega}_{c}\left(t\right)$$

where ${\tau}_{1}\left({\omega}_{0}\right)={\tau}_{0}\left({\omega}_{0}\right)+D\frac{{\lambda}_{0}^{2}}{2\pi c}\u2022{\omega}_{0}$ . Therefore, the mean squared deviation of this phase jittering can be expressed by

$${\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{\left[D\frac{{\lambda}_{0}^{2}{\omega}_{RF}}{2\pi c}-\sum _{k=1}^{n}{A}_{g,k}\frac{\mathrm{sin}\left(\frac{{\omega}_{RF}}{{T}_{g,k}}\right)}{\frac{{\omega}_{RF}}{{T}_{g,k}}}\mathrm{sin}\left(\frac{\omega +\frac{1}{2}\Delta {\omega}_{c}\left(t\right)}{{T}_{g,k}}+{\theta}_{g,k}\right)\right]}^{2}\u2022{\sigma}_{{\omega}_{c}}^{2}\}}^{\frac{1}{2}}$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}<{\{{\left[-{\tau}_{1}\left({\omega}_{0}\right)+D\frac{{\lambda}_{0}^{2}\omega}{2\pi c}-\frac{{A}_{pp}}{2}\right]}^{2}\u2022{\sigma}_{{\omega}_{RF}}^{2}+{\left[D\frac{{\lambda}_{0}^{2}{\omega}_{RF}}{2\pi c}-\frac{{A}_{\mathrm{pp}}}{2}\right]}^{2}\u2022{\sigma}_{{\omega}_{c}}^{2}\}}^{\frac{1}{2}}$$

where
${\sigma}_{{\omega}_{RF}}$
and
${\sigma}_{{\omega}_{c}}$
represents the standard deviation of Δ*ω _{RF}*(

*t*) and Δ

*ω*(

_{c}*t*), respectively.

*A*represents the maximum peak-peak amplitude of GDR. The value of ${\sigma}_{{\omega}_{RF}}$ is decided by the phase noise of RF signal, which can be expressed as

_{pp}where
${\left(\frac{N}{C}\right)}_{{f}_{c}}$
represents the phase noise value at specified frequency offset, *f _{c}*. In practice, a typical phase noise value of RF signal source is much less than -80dBc/Hz at 10 KHz frequency offset. Then
${\sigma}_{{\omega}_{RF}}$
is much lower than 300Hz, which is much less than the frequency shift induced from optical wavelength instability. So we can neglect the impact on phase deviation of the received RF signal introduced by phase noise of RF signal source.

The phase jittering will result in the increase of the mean squared sidelobe level in an N-element linear OBFN system with element spacing on half of the RF wavelength, which is one of important parameters in many phased array applications. When only considering the phase error variance, the mean squared sidelobe level can be expressed as

where *G _{0}* represents the unit gain. Therefore the larger phase jittering will result in the larger impact on the mean squared sidelobe level.

The shift of beam-pointing angle (*θ _{B}*) corresponding to the main lobe of the array antenna can be expressed in terms of TTD based on CFG as follows:

where *d* is element spacing that is half of the maximal RF wavelength for practical applications, and Δ*ω* is the optical angle frequency spacing between the inter-element arrays. Eq. (8) states that the beam-pointing shift is caused by the combination of the instability of optical sources and the GDR of CFG, which is closely correlated with *A _{g,k}, T_{g,k}* and Δ

*ω*.

## 3. Experimental results and discussion

In order to validate the impact of GDR, we did the preliminary experiments using the setup of one wavelength channel of the N-element beamformer, as shown in Fig. 1. The light from the tunable laser (Agilent-8164A with the optical-module-81680A) was externally modulated by a Mach-Zehnder modulator driven by RF signal, and then launched into an optical circulator followed by a CFG. The reflected light from the CFG fed into an N channel detection array through a 1:N demultiplexer. The delayed signals at one of channels were then detected by high-speed PDs with 3dB bandwidth of 20GHz. The amplitude and phase of the detected RF signal are measured by a network analyzer (Agilent E8364B), and hence the corresponding beamforming performance was analyzed.

Figure 2 shows a zoom of the measured GDR and the simulated model in which the simulated parameters of GDR are obtained from the Fourier transform of a measured GDR. Results describe that the measured ripples are less than 10ps (typical peak-peak value) within 3dB reflection bandwidth and the value from simulated model is similar to the practical measured GDR. So the simulated parameters of GDR can be considered as the practice parameters of GDR to analyze the performance of CFG based OBFN systems.

In the experiments, we used two CFGs to comparatively validate the effects of GDR on OBFN in different RF frequency ranges. Their peak-to-peak GDR are about 10- and 30-ps respectively and dispersion coefficients are about 51-and 212-ps/nm respectively. The standard deviation of optical source instable fluctuation is about 1.5pm according to its specification. The measured phase jittering of the detected RF signal is shown in Fig. 3. It can be noted that the phase jittering increases with the RF frequency or the amplitude of GDR, and the measured results fit well with the calculated result by Eq. (5). Therefore, we can analyze and predict the performance of N-element OBFN based on the theoretical model proposed in Section 2.

The impact of GDR on the mean squared sidelobe level in an N-element linear OBFN system is shown in Fig. 4. Given that the number of elements of the array N was 8 and *G*
_{0} was set as 1. It can be seen that the mean squared sidelobe level increases with the peak-peak amplitude of GDR and higher operation RF frequency, which is mainly derived from the larger peak-peak amplitude of GDR, the instability of optical sources and the phase noise of RF signal. All of these would induce larger phase jittering of detected RF signal at antenna element.

The calculated 8-element antenna radiation patterns of CFG based OFBNs are shown in Fig. 5(a). In our calculation, the frequency of the RF signal was set at 10GHz and the GDR parameters were set as *A _{pp}* ~30ps and the main

*T*

_{g,k}~11GHz according to measured data of one CFG. It can be seen that the beam-pointing direction depends on changing the wavelength spacing of all tunable lasers simultaneously. Since in such OBFN systems, optical wavelength of each tunable laser will corresponds to an array element. When we equally increase or decrease wavelength step of tunable lasers, different time delay interval of RF signal between the inter-element arrays will be obtained, which will determine the beam-pointing direction. For example, when the wavelength step is set as 0-, 504- and 864-pm respectively, the resulting beam-pointing direction is 0°, -30° and -60° respectively. Fig. 5(b) shows the relationship of the beam-pointing shifts and the wavelength steps for a given parameters of GDR. As shown, the beam-pointing shifts resulted from GDR periodically change with the wavelength step. Such periodical changes arise from the period of grating ripple. Moreover, the tolerable GDR of CFG could be estimated by analyzing the delay time error at the larger scanning angles.

The impact given by the GDR of CFG on the beam-pointing shift is shown in Fig. 6. The beam-pointing direction was set as 60 degree. Results show that the period of GDR can produce a dominating effect on the beam-pointing shift, which shows periodically changes with the main period of GDR. When the main period of GDR are located at some rational frequencies, the impact of GDR on the beam-pointing shift would be more serious. So these rational frequencies should be avoided when we choose the parameters of GDR. Meanwhile, when the main frequency period of GDR is smaller than the RF signal frequency, the impact resulted from the peak-peak values of GDR can be limited. Such a modelling can be used to design the CFG to meet the requirement of OFBNs application. For example, for a 10GHz OFBN system, when the beampointing error is required to be within 1° over ±60° scanning ranges, the GDR should have a period within 11GHz or a peak-to-peak amplitude within 6ps is required.

## 4. Conclusion

We investigated the impairments on optical beamforming networks given by the GDR of CFG in this paper. The influences on the RF signal phase jittering and the beampointing error of OFBN coming from the instability of optical sources through GDR are analyzed and validated experimentally. Theoretical analysis indicates the GDR can induce a dominating effect on the beam-pointing shift. This work provides a new avenue to analyze the performance of CFG-based OBFN systems, design the parameters and choose the proper chirped fiber grating for system applications..

## Acknowledgments

This work is supported in part by National Nature Science Foundation of China (NSFC) under grant No. 6052130298 and 60432020, 863 Project under grant No.2006AA01Z261, 973 Project under grant No.2006CB302805 and Project iCHIP financed by Italian Ministry of Foreign Affairs.

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