## Abstract

A cross-referenced and deadband-free method with photonic harmonic down-conversion is proposed for microwave frequency measurement based on cascaded-four-wave-mixing (CFWM) in semiconductor optical amplifiers. The proposed method enables ultra-wide and accurate frequency measurement with low-frequency spectrum detection, and at the same time achieves deadband-free and multi-tone frequency measurement by cross-referenced frequency discrimination. For a proof of concept, microwave signal measurement is experimentally demonstrated up to the 40 GHz frequency range with an 0.2-MHz measurement error. The frequency measurement features ease of configuration by only changing the low-frequency electrical local oscillators of the CFWM-based photonic harmonic down-converter.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microwave frequency measurement (MFM) is to estimate frequencies of intercepted microwave signals, which is critical to modern military and civil radio-frequency (RF) systems, such as wireless communications, electronic countermeasure (ECM), radar warning and electronic intelligence systems [1–3]. With the development of modern radar warning receivers and electronic warfare systems, MFM based on conventional electronic techniques might be ineffective for the real-time frequency estimation of the intercepted signals ranging from subgigahertz to millimeter wave [3]. In contrast, photonic-assisted MFM has attracted significant interest thanks to large instantaneous bandwidth, light weight, low loss and immunity to electromagnetic interference [4,5].

Many photonic-assisted methods for MFM had been proposed by establishing an amplitude comparison function (ACF) to map the unknown microwave frequency to the optical or electrical power ratio (i.e. frequency-to-optical power ratio mapping [6–10] and frequency-to-electrical power ratio mapping [11–15]). The MFM methods based on frequency-to-optical power ratio mapping enable wideband frequency measurement with DC power monitoring by using a pair of complementary optical filters. However, the optical ﬁltering shape should be characterized and calibrated after every tuning, and the wavelength of optical carrier should be re-aligned to the peak-null point of the optical filter, otherwise, the measurement accuracy will be affected by the environmental fluctuation [8]. The MFM method based on frequency-to-electrical power ratio mapping can be realized with two stable and complementary dispersion-induced power fading functions, which enables tunable measurement range with a measurement accuracy as high as ± 0.2 GHz. However, two sets of high-speed photodetectors are needed for wide frequency range measurement. The difficulty lies in the trade-off between the frequency range and the frequency accuracy in the frequency-to-electrical power ratio mapping [12]. Besides, other power fading-based methods can also be used to realize MFM by employing such as the cascaded MZMs [16,17], four-wave mixing [18,19] and optical linear filter [20]. The trade-off problem between frequency range and frequency accuracy still exists as well as the requirement of wideband photodetection.

Recently, a microwave down-conversion method is proposed for MFM by using two closely-spaced optical frequency combs (OFCs), which enables both ultra-wide and high-resolution frequency estimation from the minimum frequency component down-converted by each OFC [21,22]. However, the parallel-referenced frequency discrimination will bring ambiguity in retrieving the nearest comb line of the microwave signal and will lead to a number of frequency dead bands in which microwave frequencies cannot be unequivocally estimated, because different microwave frequencies corresponds to the same down-converted tones [22]. In order to overcome the dead bands, we proposed the frequency measurement by using triple closely-spaced OFCs, in which every pair of OFCs are parameterized and configured without any overlapped deadband [23]. Nevertheless, any repetition-frequency drifting of the passive mode-locked OFCs must be monitored and corrected in real-time, or it would result in mistake or even failure of deadband-free operation. Besides, the triple closely-spaced OFCs for microwave down-conversion would make the whole MFM system complex and redundant [24].

In this paper, we propose a photonic-assisted method based on cascaded-four-wave-mixing (CFWM) in semiconductor optical amplifiers (SOAs), which enables both ultra-wide and deadband-free microwave frequency measurement by cross-referencing two pairs of harmonic down-conversion tones. The method consists of two CFWM-based optical harmonic intensifiers (OHIs) and an electro-optic intensity modulator (IM) in a nested Mach-Zehnder interferometer (NMZI). The CFWM in SOAs triggers high-order harmonics generation of electrical local oscillators (LOs) in the optical domain, and enables ultra-wide harmonic down-conversion of microwave signals under test (SUT) in the electrical domain. The microwave frequency is therefore unequivocally determined by cross-referencing two pairs of harmonic down-converted tones within the LO frequencies. The proposed method can be operated in an ultra-wide measuring frequency range through high-order harmonic down-conversion of the LO signals, and configured with flexible measuring frequency range by adjusting the electrical LO frequencies. In the proof-of-concept experiment, both deadband-free and multi-tone frequency measurement is demonstrated with measurement error of less than 0.2-MHz for microwave signals up to 40 GHz by utilizing two CFWM-based harmonic down-converters at the LO frequency of 1.51 GHz and 1.533 GHz. Compared with the dual-comb-based methods, our method here enables deadband-free measuring frequency range by cross-referencing the harmonic down-conversion pairs. In contrast to the triple-comb-based methods, ours achieves deadband-free operation only with dual harmonic down-conversion and features the drifting-free reconfiguration by adjusting the electrical LO frequencies.

## 2. Operation principle

As shown in Fig. 1, the proposed scheme consists of two optical harmonic intensifiers (OHI1 and OHI2) and an optical IM in a NMZI. In the upper and lower arm of NMZI, each OHI consists of an optical intensity modulator (IM-1 or IM-2) and a semiconductor optical amplifier (SOA-1 or SOA-2), which is used to carry the electrical LO and intensify the high-order optical harmonics through the cascaded four-wave mixing of SOAs, respectively. In the middle arm of NMZI, the IM-3 is modulated by the intercepted microwave SUT for down-conversion with the optical harmonics from the OHIs. The harmonic down-converted tones are electrically coupled in a low-frequency balanced photodetector (BPD) and sampled by an analog to digital converter (ADC). Finally, the digital data are sent to the digital signal processor (DSP) for the low-frequency spectrum analysis, and then the frequency discrimination of microwave SUT is recovered through cross-referencing two characteristic pairs of harmonic down-converted tones. For example, if the OHIs are driven by the electrical LO at *f _{r}*

_{1}and

*f*

_{r}_{2}, the harmonic down-converted products can be paired at

*f*= |

_{a}*f*-

_{s}*nf*

_{r}_{1}|,

*f*= |(

_{b}*n*+ 1)

*f*

_{r}_{1}-

*f*|,

_{s}*f*= |

_{a}'*f*-

_{s}*nf*

_{r}_{2}| and

*f*= |(

_{b}'*n*+ 1)

*f*

_{r}_{2}-

*f*| (

_{s}*f*,

_{a}*f*<

_{b}*f*

_{r}_{1};

*f*,

_{a}'*f*<

_{b}'*f*

_{r}_{2}) with the help of

*f*+

_{a}*f*=

_{b}*f*

_{r}_{1}and

*f*+

_{a}'*f*=

_{b}'*f*

_{r}_{2}, when the SUT is assumed at the frequency

*f*Note that the two LO frequencies are set very close but slightly detuned to each other (

_{s}.*f*

_{r}_{1}≈

*f*

_{r}_{2}and

*f*

_{r}_{2}-

*f*

_{r}_{1}= Δ

*f*>0) for ultra-wide frequency range operation.

An optical carrier at the frequency of *f*_{0} is sent to the NMZI, where the electrical LO-1 at the frequency of *f _{r}*

_{1}is converted to optical domain through IM-1 in OHI1, which can be written by [25]

*B*and

_{q}*ξ*are the amplitude and phase of the

_{q}*q*th-order sideband, respectively.

*N*

_{1}is an integer representing the effective order of the harmonic sidebands with respect to the optical carrier. The modulated optical signal from IM-1 is then sent to SOA-1 in the OHI1, in which any two optical sidebands will generate new optical harmonic sidebands through four-wave mixing (FWM) effect of SOA. The newly formed sidebands can in turn interact with each other to further generate new higher-order ones through FWM, involving a cascaded process known as CFWM. The new

*n*th-order optical harmonic sideband can be written as [26]

*χ*(Δ

*f*) is the relative conversion efficiency of the FWM effect and inversely proportional to Δ

_{pq}*f*(Δ

_{pq}*f*= (

_{pq}*p*-

*q*)*

*f*

_{r}_{1}, |

*p*|≥|

*q*| and

*q*= 2

*p*-

*n*). We assume the intensified optical harmonic signal from OHI1 can be simplified as [27]

*F*and

_{n}*ϕ*are the amplitude and phase of the new

_{n}*n*th-order harmonic sideband, respectively.

*N*

_{2}corresponds to the effective order of the harmonic sidebands with respect to the optical carrier. Similarly, the intensified optical harmonic signal from OHI2 can be written by

The microwave SUT at *f _{s}* is collected by IM-3 and combined with the LO harmonics at

*nf*

_{r}_{1}and

*nf*

_{r}_{2}in the optical domain, and then detected by a low-frequency balanced photodetector (BPD). After photodetection, the optical beating signals at

*f*

_{M}_{1}= |

*f*-

_{s}*nf*

_{r}_{1}| and

*f*

_{M}_{2}= |

*f*-

_{s}*nf*

_{r}_{2}| are generated by the harmonic down-conversion.

From Fig. 1, the harmonic down-converted tones can be paired at *f _{a}* = |

*f*-

_{s}*nf*

_{r}_{1}|,

*f*= |(

_{b}*n*+ 1)

*f*

_{r}_{1}-

*f*|,

_{s}*f*= |

_{a}'*f*-

_{s}*nf*

_{r}_{2}| and

*f*= |(

_{b}'*n*+ 1)

*f*

_{r}_{2}-

*f*| in the frequency range

_{s}*f*

_{r}_{1}and

*f*

_{r}_{2}with the help of

*f*+

_{a}*f*=

_{b}*f*

_{r}_{1}and

*f*+

_{a}'*f*=

_{b}'*f*

_{r}_{2}. In our method, the frequency discrimination is divided into two frequency ranges (range I and II) for cross-referencing the harmonic down-converted tones, according to the relative frequencies of

*f*,

_{s}*nf*

_{r}_{1}and

*nf*

_{r}_{2}. In the case of range I, that is,

*nf*

_{r}_{1}<

*f*<

_{s}*nf*

_{r}_{2}(

*f*

_{r}_{1}<

*f*

_{r}_{2}), as shown in Fig. 1, the four harmonic down-converted tones at

*f*,

_{a}*f*,

_{b}*f*and

_{a}'*f*can be expressed as

_{b}'*n*can be determined by

In the case of range II, that is, *nf _{r}*

_{2}<

*f*<(

_{s}*n +*1)

*f*

_{r}_{1}, the four harmonic down-converted tones at

*f*,

_{a}*f*,

_{b}*f*and

_{a}'*f*can be written as

_{b}'*n*can be determined by

With the harmonic order *n*, the microwave SUT can be accordingly recovered by

From Eq. (9), one can see that it is critical to determine the frequency range of SUT with respect to the LO harmonics, i.e. harmonic order *n* and down-converted tone *f _{a}*. In order to eliminate ambiguity, we propose here a cross-reference algorithm for the microwave frequency discrimination, as is illustrated in Fig. 2. The procedure can be described in detail as following. Firstly, through ADC and fast Fourier transform (FFT), all the harmonic down-converted tones within the frequency

*f*

_{r}_{2}are collected with the constraint condition of

*P*(

*f*)≥

_{i}*P*, in which

_{th}*P*(

*f*) is the electrical power of the down-converted component at

_{i}*f*and

_{i}*P*is the pre-set power threshold and leveled at least above the noise floor and the harmonic mixing spur of SUT. Secondly, one matrix of [

_{th}*f*,

_{i}*f*,

_{j}*f*,

_{k}*f*] is built according to the relationships of

_{l}*f*

_{r}_{1}=

*f*+

_{i}*f*(

_{j}*i*≤

*j*) and

*f*

_{r}_{2}=

*f*+

_{k}*f*(

_{l}*l*≤

*k*) to determine the generation source of the harmonic down-converted components. Then, another two matrixes of A = [

*a*

_{xy_}_{N}|

*a*

_{xy_}_{N}= ||

*f*-

_{x}*f*|-N × Δ

_{y}*f*|,

*x*=

*i*,

*j*,

*y*=

*k*,

*l*, 0≤N≤N

_{MAX}] and B = [

*b*

_{xy_}_{N}|

*b*

_{xy_}_{N}= |

*f*+

_{x}*f*-N × Δ

_{y}*f*|,

*x*=

*i*,

*j*,

*y*=

*k*,

*l*, 0≤N≤N

_{MAX}] are built based on Eqs. (8) and (6), respectively, in which N

_{MAX}= round(

*f*

_{r}_{1}/Δ

*f*) is the maximum positive integer. Through seeking the integer numbers N

*and N*

_{i}*(N*

_{j}*= N*

_{j}*+ 1), the parameters*

_{i}*n*and

*f*in Eq. (9) can be determined if both of the relationships

_{a}*a*

_{iy_}_{N}

*≤*

_{i}*ε*and

*a*

_{jy_}_{N}

*≤*

_{j}*ε*are satisfied.

*ε*is a small enough value to obtain parameter

*n =*N

*. Otherwise, through seeking the integer numbers N*

_{i}*to satisfy the relationship of*

_{i}*b*

_{iy_}_{N}

*≤*

_{i}*ε*. Finally, the parameters

*n*and

*f*are obtained as N

_{a}*and*

_{i}*f*, respectively. The microwave frequency can be discriminated with the help of Eq. (9). It should be noted that

_{i}*ε*is dependent on the finite frequency resolution of ADC. The frequency measurement accuracy depends on the sampling sequence length and the sampling rates of ADC, according to Nyquist theorem.

In our method, the ultra-wide measuring frequency range of *f _{r}*

_{1}×

*f*

_{r}_{2}/(2Δ

*f*) can be achieved when the LO frequencies

*f*

_{r}_{1}and

*f*

_{r}_{2}are set to be several orders of magnitude of their frequency difference Δ

*f*. In the conventional dual-comb-based methods, there will be ambiguity in frequency discrimination, when the frequency of SUT is located between

*nf*

_{r}_{1}+

*f*

_{r}_{1}/2 and

*nf*

_{r}_{2}+

*f*

_{r}_{2}/2, resulting in a deadband range of (

*n*

^{2}

*+ n/*2)(

*f*

_{r}_{2}-

*f*

_{r}_{1}) [22]. In contrast, our method eliminates the deadband by the cross-referenced frequency discrimination. Furthermore, when the microwave SUT includes different frequency components, every component will correspond to its separate parameter of

*n*and

*f*in the cross-reference frequency discrimination, which means that our method is applicable for measuring both single- and multi-tone SUT.

_{a}## 3. Experiment and discussion

A proof-of-concept microwave frequency measurement is demonstrated in our experiment. The optical carrier originating from a distributed feedback laser diode (DFB-LD) with a wavelength of 1550.12 nm and an optical power of 11.6 dBm is split into the NMZI. Two OHIs are located in the upper and lower arm of NMZI, respectively, which consists of an IM cascaded with a SOA. The IMs are from EOspace (AX-0S5-10) and driven by electrical LO signals at *f _{r}*

_{1}= 1.51 GHz and

*f*

_{r}_{2}= 1.533 GHz with the power of 25.6 dBm, while the SOAs are from Kamelian (SOA-NL) for optical harmonic intensification based on CFWM [28]. A wideband Mach-Zehnder modulator from Sumitomo (T·DKH1.5) is located in the middle arm of NMZI to carry the microwave SUT. The coupled signals are sent to the two ports of BPD for harmonic down-conversion. The spectrum analysis is performed with an electronic spectrum analyzer (ESA) to analogy with the ADC and FFT processing in the demonstration.

In order to demonstrate the CFWM-based optical harmonic intensification, we set the LO frequency of OHI at 1.51 GHz, 4 GHz, 6 GHz, 8 GHz, 10 GHz and 12 GHz and observe the optical spectrum after OHI with an optical spectrum analyzer (OSA, AQ6370C), respectively. As shown in Fig. 3, the CFWM-based optical harmonic intensification can also be obtained by changing the LO frequency. Therefore, the flexible and wideband harmonic sidebands can be intensified for high-order harmonic down-conversion of microwave SUT. It should be noted that the optical comb spectral lines cannot be fully resolved in the case of LO frequency 1.51 GHz due to the limited resolution of OSA (about 1.25 GHz at 1550 nm), but the wide spectral envelop will be qualified for a wideband MFM. In our experiment, two LOs are set with the power of 25.6 dBm at the frequencies of *f _{r}*

_{1}= 1.51 GHz and

*f*

_{r}_{2}= 1.533 GHz, respectively. The SOA-1 and SOA-2 are biased at 200 mA and with input optical power of −4.14 dBm and −4.51 dBm, respectively. Two intensified optical harmonic signals with different frequency spacings of

*f*

_{r}_{1}= 1.51 GHz and

*f*

_{r}_{2}= 1.533 GHz are obtained from the outputs of SOA-1 and SOA-2, respectively. In our measurement, a two-tone microwave signal is used as the microwave SUT, which is at the frequencies of 24.5 GHz (16 ×

*f*

_{r}_{1}<

*f*

_{s1}<16 ×

*f*

_{r}_{2}) and 25 GHz (16 ×

*f*

_{r}_{2}<

*f*

_{s2}<17 ×

*f*

_{r}_{1}&16.5 ×

*f*

_{r}_{1}<

*f*

_{s2}<16.5 ×

*f*

_{r}_{2}) with the power of 10 dBm. The two-tone microwave signal is harmonic down-converted by the intensified optical harmonics from OHI1 and OHI2, respectively.

Figure 4 shows the electrical spectrum of the harmonic down-converted signals within 1.533 GHz at the resolution bandwidth (RBW) of 0.2 MHz. The power threshold *P _{th}* and

*ε*are set as −50 dBm and 0.5 MHz, respectively. From the eight down-converted tones, there are four groups of down-converted tones [

*f*,

_{i}*f*,

_{j}*f*,

_{k}*f*] built with the help of the relationships of

_{l}*f*

_{r}_{1}=

*f*+

_{i}*f*(

_{j}*i*≤

*j*) and

*f*

_{r}_{2}=

*f*+

_{k}*f*(

_{l}*l*≤

*k*), as shown in Table 1. According to the proposed cross-referenced algorithm of frequency discrimination, four down-converted tones with the frequencies of 0.3398 GHz, 1.1698 GHz, 0.0278 GHz and 1.5050 GHz satisfy the frequency relationship of |

*f*+

_{a}*f*-N

_{c}*× Δ*

_{i}*f*|<

*ε*when the natural number N

*is 16. Then, the four down-converted tones for the 24.5-GHz SUT can determined as of*

_{i}*f*= 0.3398 GHz,

_{a}*f*= 1.1698 GHz,

_{b}*f*= 0.0278 GHz and

_{a}'*f*= 1.5050 GHz, respectively, and the parameters are determined as

_{b}'*n*= 16 and

*f*= 0.3398 GHz, from which the SUT frequency

_{a}*f*

_{s}_{1}is calculated as 24.4498 GHz with an error of 0.2-MHz.

For measuring the SUT frequency *f _{s}*

_{2}, the frequency relationships of ||

*f*-

_{a}*f*|-N

_{c}*× Δ*

_{i}*f*|<

*ε*and ||

*f*-

_{b}*f*|-N

_{d}*× Δ*

_{j}*f*|<

*ε*are satisfied with the natural numbers N

*= 16 and N*

_{i}*= 17. The four down-converted tones of*

_{j}*f*,

_{a}*f*,

_{b}*f*and

_{a}'*f*for the 25-GHz SUT can determined as 0.8398 GHz, 0.6700 GHz, 0.4720, and 1.0608 GHz, respectively. The SUT frequency

_{b}'*f*

_{s}_{2}is therefore calculated as 24.9998 GHz with

*n*= 16 and

*f*= 0.8398 GHz. It is worth noticing that the microwave SUT at

_{a}*f*

_{s}_{2}= 25 GHz will fall into the measurement deadband of

*nf*

_{r}_{1}+

*f*

_{r}_{1}/2<

*f*<(

_{s}*n*+ 1)

*f*

_{r}_{2}-

*f*

_{r}_{2}/2 (i.e. 24.915 GHz<25 GHz<25.2945 GHz) and results in measurement ambiguity by using the conventional dual-comb-based method. It is obvious that the cross-reference frequency discrimination enables deadband-free microwave frequency measurement in our method. Moreover, all the down-converted tones show extremely narrow spectrum lines due to the inherent coherence of the SUT-modulated and harmonic-intensified optical signals originating from the same optical carrier, which indicates that the frequency measurement is robust to the wavelength drifting and linewidth of the optical source. Therefore, a deadband-free frequency measurement can be expected for discriminating the wide-band and multi-tone SUT with low-frequency and high-resolution ADC.

In Fig. 4, there are some spur frequency components observed, which are from the beating between the high-order SUT-modulated sidebands (*p*>1) and the intensified optical harmonics. In order to evaluate the spur effect, the spur-free dynamic range (SFDR) is measured by applying two-tone RF signals at 15.1 GHz and 15.5 GHz. The measured fundamental down-converted tone and third-order intermodulation distortions (IMD3) as a function of the input RF power are shown in Fig. 5. As the noise floor is −128.52 dBm/Hz, the calculated SFDR of our system is as high as 93.52 dBc·Hz^{2/3}. Theoretically, the spur effect can be also eliminated through analyzing the power ratio between the desired components and the spur components, which are resulting from the nonlinearity of IM-3 and more than 20 dB (20**lg*[*J*_{1}(*m*)/*J*_{2}(*m*)]) in the case of 0<*m*<0.4.

The microwave frequency measurement for microwave signals up to 40 GHz is demonstrated with the corresponding measurement uncertainty. As shown in Fig. 6, the measured (circle) frequencies are agreeable with the theoretical (line) results, in which the measured error is less than 0.2 MHz. In the proof-of-concept experiment, the spectrum analysis is performed with an ESA to analogy with the ADC and FFT processing for demonstration. The measurement accuracy can be further improved by high-resolution ADC and FFT, when the signal-to-noise is guaranteed. As we know, the FFT takes a block of time-domain waveform and resolves into its frequency spectrum. The frequency resolution is determined by the acquisition time or window. According to Nyquist theorem, the frequency accuracy improves as the acquisition time or window increases. For example, an acquisition time of 5 μs corresponds to a frequency resolution of 0.2 MHz. Note that the ESA will run slower in measurement speed than ADC and FFT. Even so, the ESA only needs to scan in about 1.533 GHz for measurement up to 40 GHz. In contrast, the commercial ESA needs to scan in full frequency range (40 GHz), ours will save 96.17% (1-1.533/40 = 1-3.83%) sweeping time and the measurement will run 26 times faster in brief. Besides, as shown in Table 2, ours enables ultra-wide measurement range with narrow sweeping bandwidth (about 1.533 GHz) for the MFM up to 50.3 GHz (*f _{r}*

_{1}×

*f*

_{r}_{2}/(2Δ

*f*)), thanks to the wideband CFWM-based harmonic down-conversion. The measurement can be largely speeded up, e.g. 0.1 s for ESA and about 5 μs for ADC + FFT within 1.533 GHz, due to the reduced sweeping bandwidth, while it will take 2.5 s to take a full range frequency sweeping (40 GHz) for a commercial ESA. In principle, the frequency measurement accuracy of the proposed method can be improved through increasing acquisition time or sampling points used for FFT calculation. In practical applications, the measurement accuracy should also be compromised with the measurement speed.

## 4. Conclusion

A cross-reference and deadband-free method with photonic harmonic down-conversion is proposed for microwave frequency measurement based on cascaded-four-wave-mixing in semiconductor optical amplifiers, which enables ultra-wide, high-accuracy and multi-tone frequency measurement with low-frequency detection and discrimination. In the proof-of-concept experiment, the multi-tone frequency measurement is demonstrated up to 40 GHz frequency range with 0.2-MHz measurement error. Unlike the ACF-based method, our method enables multi-tone frequency measurement and eliminates the trade-off between the frequency range and accuracy. Compared with the competitive method based on dual OFCs, our method avoids the limitation of deadband by the cross-referenced frequency discrimination, and at the same time it enables drifting-free and reconfigurable frequency measurement by tuning the electrical LO frequencies of the CFWM-based optical harmonic intensification.

## Funding

National Natural Science Foundation of China (NSFC) (61875240 and 6192780120); National Key Research and Development Plan (2018YFE0201900, 2018YFB2200700); The Joint Research Fund of MOE (6141A02022436); Innovation Special Zone Funds (18-163-00-TS-004-040-01).

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