The evaluation of the parameters of radio-frequency (RF) signals with high accuracy, particularly in high-frequency regions, is in high demand for application in communication and measurement based on radio detecting and ranging (radar). Conventionally, RF signal characterization relies on nonlinear electronic devices affected by electromagnetic noise; however, the measurement accuracy would be hardly improved further, and it has a tendency to degrade with an increasing frequency of RF signals. Then for improving the accuracy in RF signal evaluation, the approach being free from such RF devices would be useful. To overcome these drawbacks, microwave/millimeter-wave photonics seems to be a promising solution owing to some features of the lightwave such as broad bandwidth and low-loss transmission [1], and it has been applied in the phase adjustment of an RF signal [2,3], extreme narrow-bandwidth RF signal filtering [4–6], frequency upconversion [7,8]/downconversion [9,10], multiplexing/demultiplexing [11], radar [12], and high-speed waveform measurement [13]. This is assisted by highly sophisticated lightwave modulation/detection techniques that enable the adoption of a precise and advanced modulation format for lightwaves [14].

In this Letter, we propose and demonstrate a scheme for estimating RF signal parameters using phase modulation and interference of lightwaves. Phase modulation of the lightwave allows the complex amplitude of the RF signal to be reflected onto optical sidebands constituting the lightwave. Optical sidebands are evaluated by detecting the signal originating from optical interference with a second lightwave (also phase-modulated by the reference RF signal but different from the RF signal under test). Because this scheme enables wide-range frequency downconversion of the signal, wide bandwidth is not required in the photodetector and post-amplifier; thus, the gain-bandwidth product can be weighed to the gain. Moreover, this scheme does not involve the evaluation of each sideband and optical-domain filtering by using a special optical filter to selectively extract each sideband. Although RF signal phase detection has been demonstrated previously [15], it suffered from unavoidable residual measurement errors in the order of a few degrees, originating from an optical semiconductor amplifier that was used to induce four-wave mixing.

 

Fig. 1. Proposed scheme of RF signal measurement: (a) two-parallel optical phase modulator at in-phase condition driven by RF signals and a dither signal. For simplicity, ${-}{{2}^{{\rm nd}}}\sim + {{2}^{{\rm nd}}}$-order optical frequency components are shown. Typical complex optical spectra due to optical phase modulation are also shown. Note that phase of each optical sideband is determined by the product of its harmonic order and the RF signal phase, while all optical frequency components generated by the REFERENCE RF signal have no optical phase shift so that only the fluctuation due to the dither signal is superposed. (b) Optical/electrical conversion at a photodiode: each plane corresponds to that on (a).

Download Full Size | PDF

Figure 1 shows a schematic of the proposed method in which two phase-modulated lightwaves interfere. After the monochromatic lightwave is divided, two lightwaves are independently modulated in phase: one by an RF signal under test and the other by the reference RF signal whose amplitude is adequately fixed in advance. The phase modulations are performed at each arm of a Mach–Zehnder optical modulator (MZM), and the interference signal of both lightwaves is detected using a slow-response photodiode (PD), such that only low-frequency components of the interference signal are obtained. The interference signal is oscillated by superposing either of the phase-modulated signals onto a slowly oscillated bias voltage (in the frequency of a few kHz: hereafter referred to as “a dither signal”), due to the beats between the same-order optical sidebands of each phase-modulated lightwave. The frequency is downconverted to the order of the dither signal. The beat signal amplitude depends on that of each optical sideband; that is, the complex amplitude of the RF signal can be evaluated from the beat obtained from the dither signal. Moreover, this evaluation procedure does not involve a direct analysis of the original RF signal frequency band and the originally modulated lightwave in the baseband. Based on the assumption that the MZM has an infinite extinction ratio and is driven by the two sinusoidal RF signals, lightwave $ E $ immediately before the direct detection is

Based on the assumption that the sufficiently small sinusoidal phase oscillation is induced by the dither signal, ${\theta _{\rm B}}$ is expressed as

By substituting ${\theta _{\rm B}}$ into Eq. (2) and neglecting the angular frequency components of ${3}{\omega _{\rm d}}$ or above based on the Taylor expansion due to low-pass filtering by the PD, we derive the normalized intensity, $| {{E / {E_0}}} |_{{\rm LPF}}^2$, as follows:

Eq. (6) shows that the beat signal contains a ${2}{\omega _{\rm d}}$ sinusoidal-oscillation component, and its amplitude depends on ${J_0}(\Delta {\alpha ^\prime})$ and $\Delta {\theta _{\rm d}}$; that is, $\kappa$ and $\phi$, arguments of $\Delta {\alpha ^\prime}$, can be evaluated from the ${2}{\omega _{\rm d}}$-oscillation amplitude under the given value of $\Delta \alpha$, when the dependence on $\Delta {\theta _{\rm d}}$ is eliminated. For deriving Eq. (6) from Eq. (2), the Jacobi–Anger expansion slightly modified for sine function is applied, and only the zeroth term in real part remains because of the slow response of the PD.

The $\Delta {\theta _{\rm d}}$ dependence of the ${2}{\omega _{\rm d}}$ oscillation amplitude can be eliminated by experimentally evaluating the normalized intensity in advance, under the case of ${J_0}(\Delta {\alpha ^\prime}) = 1$; that is, $\Delta {\alpha ^\prime} = 0$, meaning that only the dither signal is applied to the electrode of the MZM. Based on ${A_{{\rm RF} {\text -} 0}}$ and ${A_0}$, which represent the experimentally obtained ${2}{\omega _{\rm d}}$ oscillation amplitudes when $\Delta {\alpha ^\prime}$ equals to $\Delta \alpha _{{\rm RF} {\text -} 0}^\prime $ and zero, respectively, the following relation should be satisfied for the parameters of the RF signal under test ($\kappa$ and $\phi$):

Then, $\phi$ can be obtained if $\kappa$ is given a priori, and vice versa. Even if both $\kappa$ and $\phi$ are unknown, both parameters can be obtained through additional measurements when the phase of the REFERENCE RF signal is 90°-shifted. For the additional results, the following relations are derived:

In the aforementioned procedure, $\Delta \alpha$ is used as a parameter to determine its adequate value based on the model. Figure 2 shows trajectories of the simulated direct-detection-signal amplitude on a complex plane at $\Delta \alpha = 2.405$ corresponding to the Zero of ${J_0}(x)$. $\kappa$ is assumed to be in the range from 0.02 to 0.1, considering that in many cases the RF signal to be evaluated would be weak. In the case of $\kappa = 0.1$, the driving voltage corresponds to ${0.253}\;{{V}_{0 {\text -} {\rm p}}}$, assuming that the half-wave voltage of the optical phase modulator is ${3.3}\;{{V}_{0 {\text -} {\rm p}}}$. The length from the origin to each point constituting each trajectory corresponds to the output amplitude of the detection signal. The angle against the horizontal axis indicates the phase of the RF signal under test and against the reference RF signal: the positive directions of the horizontal axis and the vertical axis correspond to the phase of 0° and 90°, respectively.

 

Fig. 2. Trajectory of direct-detection signal on the polar plane at an induced optical phase of $\Delta \alpha = 2.405$. The relative amplitude of the RF signal under test is in the range of $\kappa = {0.02 - 0.17}$. Horizontal axis direction corresponds to the phase of the reference RF signal.

Download Full Size | PDF

 

Fig. 3. (a) Time trace of dither signal (red line) and AC component of direct-detection signal at the given RF signal phase of ${-}{3}^\circ$ (thick black line). With a decrease in the line thickness and its color intensity, the given RF signal phase is shifted by ${-}{15}^\circ$. (b) ${ \cos }\phi$ and $\phi$ estimated from the results shown in Fig. 3(a).

Download Full Size | PDF

For the case of $\kappa = 0.02$ and 0.04, the trajectory shows a circle with the diameter of the two intersections on the origin and the positive horizontal axis; that is, the obtained amplitude proportional to the cosine of the RF signal phase. Further, these circles composed of each trajectory degenerate: one belongs to the first and fourth quadrants and the other to the second and third quadrants with “negative amplitude” (phase inversion of the direct-detection signal). This implies that the trajectories are rotated in a clockwise direction on the complex plane when we shift the phase of the reference RF signal. Thus, both $\phi$ and $\kappa$ can be estimated by adopting the results obtained from the two different phases of the reference RF signal—0° and 90°. Unlike this case, with an increase in $\kappa$, the degenerated circle is split, indicating that the output signal amplitude depends on the sign of ${ \cos }\phi$. However, the effect of the asymmetry on the estimation results can be compensated by comparing the phase of the direct-detection signal with that of the ${2}{\omega _{\rm d}}$ cosine waveform obtained from the dither signal to decide the sign of ${ \cos }\phi$.

A proof-of-principle experiment was conducted using an experimental setup in which an external-cavity diode laser with a wavelength of 1550 nm was used as a light source. Optical power of 3 dBm was launched into the MZM composed of two optical phase modulators [18] by using a polarization controller and an optical isolator. The MZM was adjusted to the in-phase condition by applying a bias voltage onto both its arms, and then a dither signal with a frequency of ${f_{\rm d}} = 10\;{\rm kHz}$ was input to one electrode. As well as the dither signal source used in the experiment, a typical sinusoidal source has sufficiently low background noise so that the source would not deteriorate the evaluation results. The light passing through the MZM was received by an InGaAs PD module (bandwidth, 50 kHz), and time traces were acquired using an oscilloscope. The dither signal applied into the MZM was also acquired simultaneously. Each arm of the MZM was driven using a 10 GHz RF signal. The power of the reference RF signal was adjusted to achieve an induced optical phase of 2.4. To focus on the preciseness of the RF phase estimation, the power of the other RF signal under test was assumed to be known in advance and was set to ${-}{15.34}\;{\rm dB}$ against the reference RF signal, corresponding to the condition of $\kappa = 0.170$.

Figure 3(a) shows time traces of the PD output voltage and the dither voltage, in which the horizontal axis is normalized by the period of the dither signal. The amplitude of the PD output voltage gradually decreases with an increasing phase of the RF signal, and the voltage is flipped when the RF phase shift exceeds 180°. This is consistent with the model analysis results, thereby demonstrating the validity of the proposed method and analysis model. Further, the phase of the RF signal was estimated by using the amplitude of these waveforms with Eqs. (3) and (7). The evaluation of the PD signal amplitude is based on its orthogonality with the frequency-doubled cosine waveform calculated from the applied dither signal, i.e., Fourier analysis, which also eliminates the effect of the noise. Figure 3(b) shows the estimated ${ \cos }\phi$ versus adjusted value $\phi$, based on six time-measurement results. The figure also depicts the estimated $\phi$, which is obtained from the estimated ${ \cos }\phi$ and a priori known continuously in the change of the RF phase (no jump occurs in phase). The estimated ${ \cos }\phi$ agrees well with the cosine of the given $\phi$; the highest relative error of 0.07 is observed at $\phi = 0^\circ$ because the cosine function has a local maximum at the argument. The experimentally evaluated maximum standard deviation of ${ \cos }\phi$ and $\phi$ is 0.02 and 4.1°, respectively.

In summary, we estimated an RF signal parameter using frequency downconversion via optical signal processing. Based on the model constructed, the dependence of the direct-detection signal on the complex amplitude of the RF signal was analyzed. An RF-phase-dependent low-frequency signal was obtained via direct detection by a PD, and the original phase of microwave-range RF signal was successfully estimated by using experimentally obtained data and low-frequency signal processing within a range of a few kHz. This scheme would be also useful for the characterization of RF signals in further high-frequency regions such as the millimeter-wave band.