## Abstract

We numerically investigate the operational principle and performance of stimulated Brillouin scattering based multiple microwave frequency signals measurement. The unknown signals are processed specially to generate a gain region which is measured by phase modulation to amplitude modulation converting. By sweeping the vector network analyzer, both single and multiple frequencies measurement can be achieved. The loss spectrum generated by one of the pumps is fully compensated by the gain spectrum of the other pump, which increases the measurement range from 2*ν _{B}* to 4

*ν*.

_{B}©2013 Optical Society of America

## 1. Introduction

Frequency measurement ranging from subgigahertz to millimeter (mm)-wave frequency is of great importance for the application in modern radar system and electronic warfare, which has been widely investigated in electric domain [1]. Over the past few decades, photonic assisted microwave measurement techniques have been developed due to its inherent advantages compared with electronic techniques, such as large bandwidth, light weight, small size, low loss, and immunity to electromagnetic interference [2]. Recently, a number of photonic approaches have been demonstrated to implement microwave frequency measurement. Instantaneous frequency measurement (IFM) based on microwave power or optical power comparison of two channels is realized by constructing amplitude comparison function (ACF). Several methods have been proposed to construct ACF using, such as, dispersion induced microwave or optical power penalties [3–8], polarization domain interferometer [9], and optical filter or filter subsystem [10–12]. Also an on-chip photonic IFM system has been demonstrated which is a promising solution to resist environmental disturbance [13].To avoid frequency ambiguities, a monotone interval is chosen to get a unique mapping between power ratio value and microwave frequency. A steep ACF over the entire range of measurement is required since flat ACF will increase the measurement error. Although IFM with large measurement range and high resolution can be realized, only single frequency signal measurement is accurate. Multiple frequency signals measurement which is highly desired in realistic spectrally-cluttered environment has also been demonstrated. An electrically tuned fiber Bragg grating is used to realize frequency scanning measurement ranging from 2 to 9 GHz in [14]. In [15], a photonic based channelized receiver is presented using phase-shifted chirped fiber Bragg grating and photodetector array. A system based on frequency-time mapping for two 20 GHz spaced RF signals measurement utilizing a dispersive medium is reported in [16]. A two-tone RF signals measurement is presented in [17] by processing interferogram which is measured by sweeping tunable laser, and 15MHz resolution is experimentally demonstrated. Another similar version which can realize real-time measurement with limited resolution is proposed in [18]. A method using a spectrally sliced incoherent source is proposed in [19], in which two etalons with different free space range are employed and 5 GHz channel spacing and ± 2.5 GHz accuracy are achieved. In [20], a seven bands simultaneous and parallel instantaneous frequency measurements with ± 500 MHz error over the entire 1–40 GHz band is demonstrated.

In [21], a technique combining stimulated Brillouin scattering (SBS) and phase modulation to amplitude modulation (PM to AM) conversion is proposed, and less than 30MHz measurement error has been demonstrated. But the vector network analyzer (VNA) in [21] is only used to provide probe signals; additional synchronization digital oscilloscope is needed to complete the measurement. In this paper, a novel configuration is proposed, where the VNA not only provides probe signals but also performs frequency measurement so that the oscilloscope is removed. We have theoretically analyzed the operational principle and measurement performance. By extending the scheme from one pump to two pumps, the measurement range is broadened from 2*ν _{B}* to 4

*ν*for multiple frequencies measurement.

_{B}## 2. Operational principle

The schematics of the proposed microwave frequency measurement are shown in Fig. 1. The phase modulator is driven by a sinusoidal microwave signal with scanning frequency
*f _{m}* generated by a VNA. If the phase-modulated signal is applied
directly to photodetector (PD), no signal would be detected, except a dc since the beating
between the optical carrier and the upper sideband will cancel completely the beating between
the optical carrier and the lower sideband, due to the fact that the beat signals are out of
phase with a balanced intensity.

In the proposed scheme, the unknown signal is up-converted by dual parallel Mach-Zehnder modulator (DPMZM) to generate the SBS pump and the VNA launched sweeping frequencies act as probe signals. When sidebands generated by phase modulation locate in the pump induced gain region, amplification caused by SBS process will be achieved to break the balance of sideband intensity. Hence the PM to AM conversion will be realized which will be displayed in VNA after detecting, and then the unknown frequency will be obtained since the pump is specially processed so that the frequency interval between gain region and carrier is equal to unknown frequency. The SBS process is usually described as nonlinear interaction between two counter-propagating waves via an acoustic wave. If a particular phase matching condition is satisfied (namely *f _{probe}* =

*f*-

_{pump}*ν*), fraction of pump wave is scattered off into the probe wave, so that the global effect of the SBS process demonstrates a generation of narrow-band gain resonance around the frequency of

_{B}*f*-

_{pump}*ν*, resulting in an exponential growth for the probe wave. On the contrary, the energy transfer can be regarded as the generation of a loss resonance around the frequency of

_{B}*f*+

_{pump}*ν*as far as the pump wave is concerned. Thus, the SBS interaction leads to a downshifted gain region where a counter-propagating signal is amplified and an upshifted loss region where the counter-propagating signal is attenuated.

_{B}First, only one pump is employed as is shown in Fig. 1(a).
The light source provided by laser diode (LD) with the frequency *f _{c}*
is divided into two paths. The upper path is phase-modulated by VNA launched microwave signals.
The lower path is modulated by DPMZM1 and DPMZM2 which are driven by Brillouin frequency shift

*ν*and unknown signal

_{B}*f*respectively, to generate carrier-suppressed upper sideband [22]. Therefore the final output performs as pump wave with the frequency separated by

_{x}*f*+

_{x}*ν*away from carrier. The phase-modulated signals whose states of polarization are aligned by polarization controller (PC) are sent into a dispersion shift fiber (DSF) to interact with the counter-propagating pump wave. Thus the PM to AM conversion at

_{B}*f*will be realized which will be observed by VNA after PD detecting. The spectrum processing of one-pump based frequency measurement is shown in Fig. 2(a). Both biases of DPMZM1 and DPMZM2 are controlled to obtain only the first-order upper sidebands. So the gain region is centered at

_{x}*f*to realize PM to AM conversion. Noting that another frequency centered at

_{x}*f*+ 2

_{x}*ν*also will cause PM to AM conversion because of SBS caused attenuation. When multiple frequency signals are measured simultaneously, especially when existing two frequencies separated by 2

_{B}*ν*, the high frequency signal cannot be measured correctly because the gain region corresponding to high frequency signal pump will be superimposed by low frequency signal pump induced loss region. So the measurement range is limited within 2

_{B}*ν*for simultaneous measurement of multiple frequency signals if only one pump is applied.

_{B}Second, an extended scheme based on two pumps shown in Fig. 1(b) is designed to reach wider range. When the two pumps are spectrally separated by 2*ν _{B}* with the same power, the loss spectrum of pump 1 can be fully compensated by the gain spectrum of pump 2 [23]. The global effect of gain-loss compensation manifests that the spectral space between gain and loss region is broadened from 2

*ν*to 4

_{B}*ν*, which will lead to a doubled measurement range for multiple frequency signals. The spectral process is shown in Fig. 2(b). Firstly, DPMZM1 is specially driven by 2

_{B}*ν*to generate SSB-SC modulation. And secondly, the unknown signal

_{B}*f*is injected according to DPMZM2 with the same modulation pattern. At last, the amplitude modulator is biased at the minimum transmission point to generate a double-sideband suppressed carrier (DSB-SC) modulation. For the amplitude modulator, the driven frequency is specially set to be

_{x}*ν*to generated two sidebands separated by 2

_{B}*ν*where the lower sideband is used to perform frequency measurement and the higher sideband is used to implement gain and loss compensation. Since the space between gain and loss can be expanded to 3

_{B}*ν*and 4

_{B}*ν*by introducing pump 3 and pump 4 in a similar way, it is easily inferred that there is no theoretical limitation on the range of the SBS based frequency measurement.

_{B}## 3. Numeric simulation and discussion

Under small signal modulation, only the optical carrier and the two first-order sidebands are considered, and the output optical field after the PM is given by

*f*is optical carrier frequency and

_{c}*f*is microwave modulation frequency,

_{m}*J*(•) represents the

_{n}*n*th-order Bessel function of the first kind with

*n*= 0, ± 1, and

*m*is the phase modulation index.

In this paper, the first-order upper sideband is selected to be processed by SBS. The Brillouin gain and loss can be expressed as [24]:

*g*

_{0}=

*g*∕

_{B}I_{p}L_{eff}*A*,

_{eff}*ν*denotes Brillouin frequency shift and Δ

_{B}*ν*is the Brillouin linewidth,

_{B}*f*is the frequency offset to the center of Brillouin linewidth of gain (for

*g*(

*f*)) or loss (for

*a*(

*f*)),

*g*is line center gain,

_{B}*I*is power of pump wave,

_{P}*L*and

_{eff}*A*are effective fiber length and effective mode area of DSF respectively.

_{eff}When considering SBS process, the optical field before PD can be given by:

*f*is equal to

_{p}*f*+

_{c}*ν*+

_{B}*f*according to Fig. 2, and

_{x}*f*is the unknown frequency. So Eq. (4) can be rewritten as:

_{x}Omitting the dc and the small second harmonic components, the optical power input into the PD is expressed approximately as:

In the numerical simulation, several parameters are assumed: *g*_{0} = 5,
*ν _{B}* = 10GHz, Δ

*ν*= 40MHz.The calculation results are shown in Fig. 3. Figure 3(a) demonstrates that only one single frequency signal is input to be measured at a time, and

_{B}*f*equals 1GHz﹑5GHz﹑10GHz﹑15GHz﹑25GHz﹑35GHz and 48GHz respectively. It can be seen that PM to AM conversion has occurred in both gain and loss region with permanent interval of 2

_{x}*ν*. When multiple unknown frequencies are input simultaneously,

_{B}*f*is replaced by

_{p}*f*, and

_{pk}*f*=

_{pk}*f*+

_{c}*ν*+

_{B}*f*(where

_{xk}*k*= 1, 2, 3…N, and N is total number of unknown frequencies), the optical field is given by

*f*

_{x}_{1}= 1GHz,

*f*

_{x}_{2}= 5GHz,

*f*

_{x}_{3}= 10GHz and

*f*

_{x}_{4}= 15GHz is shown in Fig. 3(b).

Noting that when multiple signals are input simultaneously, especially when existing two
frequencies *f _{x}*

_{1}and

*f*

_{x}_{2}, where

*f*

_{x}_{2}=

*f*

_{x}_{1}+ 2

*ν*,

_{B}*f*

_{x}_{2}cannot be observed because of overlapping of gain and loss region as is shown in Fig. 4(a), but it can be detected if only

*f*

_{x}_{2}is measured shown in Fig. 4(b) as a comparison. Thus the measurement range of multiple frequencies is limited within 2

*ν*.

_{B}Then two pumps with spectral space of 2*ν _{B}* are designed to broaden the available measurement range. The optical field before PD is written as:

The numeric results of single frequency measurement based on two pumps are shown in Fig. 5(a), where the interval between gain and loss has been
doubled. The multi-frequency measurement at frequencies of
*f _{x}*

_{1}= 5GHz,

*f*

_{x}_{2}= 15GHz,

*f*

_{x}_{3}= 25GHz and

*f*

_{x}_{4}= 35GHz is shown in Fig. 5(b), and the available measurement range increases to 4

*ν*.

_{B}The accuracy for our proposed scheme depends on both the step of frequency-scanning of VNA and
the principle used to realize frequency measurement. In principle, the frequency information of
the microwave signal depends on the Brillouin frequency shift
*ν _{B}*, which is related to pump wavelength, surrounding
temperature and the applied strain on the fiber, and it can be given as:

*n*is the refractive index,

*V*is the speed of acoustic wave and

_{a}*λ*is the wavelength of pump light. So,

_{p}*Δ*T and strain

*Δε*, which can be written as [25]

*C*is the temperature coefficient and

_{T}*C*is the strain coefficient for the fiber, and

_{ε}*C*is near 1MHz/°C and

_{T}*C*is near 0.05MHz/με. In practical frequency measurement, the strain is constant but the surrounding temperature will fluctuate leading to the deviation of

_{ε}*ν*, so the temperature control is required. The last term on the right-hand side of Eq. (2) is rewritten as:It is caused by wavelengths variation of pumps which is inevitable for our proposed frequency measurement system because the pumps are generated by injecting the unknown frequencies with SSB-SC, which means that the pump wavelengths will change with different incoming unknown frequencies; hence the measurement error is introduced. In our proposed scheme,

_{B}*λ*is the carrier at 1.55μm and the corresponding

_{p}*ν*is 10GHz, and the relationship between

_{B}*Δν*

_{B}_{2}and

*Δλ*is shown in Fig. 6. For our proposed one pump and two pumps scheme, the measurement range is from 0 to 20GHz and 40GHz, so the corresponding

_{p}*Δλ*varies from 80pm to 240pm and 400pm, which means the maximum error is 1.54MHz and 2.58MHz. Therefore, if the temperature fluctuation is controlled within 0.05°C and the strain is kept to be a constant, the surrounding fluctuation caused error can be neglected [26]. If the scanning step of VNA is less than the maximum error caused by the variation of

_{p}*ν*, the measurement accuracy for the proposed system is less than ± 3MHz, or the accuracy is determined by the scanning step of VNA.

_{B}For simultaneous measurement of multiple frequency signals the distinguishable frequency interval is crucial. When two frequencies of 10 GHz and 10.036GHz are input at different moment, they can be identified easily as is shown in Fig. 7(a). But if they are input simultaneously, the result indicates that the frequency peak has deviated. The reason is that the space between the two measured frequencies is too close which leads to the overlap of these adjacent gain regions. Moreover, identification for those frequencies will be a difficulty when they get close too much. In order to distinguish the adjacent unknown frequencies, here it is assumed that the valley amplitude shown in Fig. 7(b) must be 3dB lower than both of the two peaks. By calculating, the valley amplitude reduces with the increase of frequency space because of the diminishment of overlapping as is shown in Fig. 8(a). The minimum frequency space of 35.8MHz corresponding to 3dB valley is achieved. Meanwhile, the relationship between frequency offset and the frequency space is given in Fig. 8(b), and the maximum offset is 2.3MHz at the space of 35.8MHz. However, when the two unknown frequencies are separated by more than 120MHz, the frequency offset can be neglected. So when multiple simultaneous signals are measured, additional measurement error caused by overlapping the adjacent frequencies must be considered if they get too close. The measurable frequency number is determined by the ratio of the maximum measurement range to the minimum distinguishable frequency interval. Although large measurement range and high accuracy for multiple frequency signals measurement can be implemented, high resolution frequency sweeping over the whole measurement range will be time-consumption. The tradeoff between measurement range and accuracy must be made to get the frequency measurement close to real time.

## 4. Conclusion

Single and multiple frequency signals measurement based on SBS in combination with PM to AM conversion has been proposed and analyzed. Compared with one pump scheme, measurement range for MFSM has been broadened from 2*ν _{B}* to 4

*ν*when two pumps are applied. The measurement error less than 10 MHz has been obtained. The minimum frequency space of 35.8MHz with 2.3MHz frequency offset has been get for simultaneous signals measurement. Although frequency sweeping is adopted, it is still a promising solution for frequency measurement with large range, high accuracy and high real-time.

_{B}## Acknowledgments

The authors are grateful to Science and Technology Development Plan of Jilin Province (Grant Nos.20110314, 20120324), the National Natural Science Foundation of China (Grant Nos.61077046, 61274068, 61275035) for the support in the work, and Chinese National Programs for High Technology Research and Development (Grant No.2013AA030902).

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