A readout using a frequency-chirped laser is investigated in a spectrum analyzer with spectral-hole-burning (SHB). An analysis based on the Bloch equations is presented for the spectral distortion due to a fast readout, and a recovery algorithm is developed for the distortion. The experiment of a SHB spectrum analyzer is executed to demonstrate the optical spectral distortion due to fast readout. The experimental spectral distortion is recovered by the recovery algorithm developed from the Bloch equations.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
A spectrum analyzer based a SHB material has received increased attention for radio-frequency (RF) signal processing due to large dynamic range, fine spectral resolution and broad bandwidth over traditional methods using a frequency-chirped laser readout [1–8]. In which, the modulated optical signal is exposed to a SHB material, the power spectrum of the input signal will be stored in the material temporarily. Consequently, the optically carried RF spectrum is recorded in the material absorption bandwidth. The lifetime of this spectral photograph depends on the population relaxation time of the system. During this lifetime, the spectrum is read out with a chirped laser, whose frequency is scanned over the whole absorption profile [7-8]. A faster readout of the signals is always desired for the RF spectrum analyzer, however, this will lead to the spectral distortion [9–11]. A perturbation theory [7–9] has been introduced to investigate and recover the spectral distortion. One can explain all the process of the spectrum analysis using spectral-hole-burning in one theory. However, the theory does not give more insight into the spectral distortion.
In this manuscript, a theory based on Bloch Equations is developed to describe the he spectral distortion of SHB materials spectrum analyzer, the experiments are carried out to demonstrate the distortion due to the fast readout chirp. The distortion of the experimental results is recovered by a method from the Bloch Equations.
2. Theory and analysis
The frequency-chirped read out is used to create a power spectrum of the input signals in the spectrum analyzer based a SHB material. In order to have a Lorentzian spectrum, the read out chirp rate should be limited by the spectral resolution as the Eq. of κ<<Γ2, where κ = B/t is the chirp rate with a chirp bandwidth B and time t, Γ is the spectral resolution . If the chirp velocity is too fast to break the Eq., the readout will produce a distorted spectrum.
Actually, the fast-chirped readout can be treated by the interaction of the classical coherent electromagnetic with a two level quantum system governed by Bloch Equations, where the electric field acts as a source to drive the atomic dipoles. Then, one can describe the fast-chirped readout phenomenon by solving the Bloch Equations. The well-known optical Bloch Equations  for two level atoms are shown as:
In our discussion, the frequency sweep is supposed to be continuous over the two level resonance and fast enough across the spectrum. Then, the population difference is considered to be invariant and can be replaced by its equilibrium value of W0.
Combining the first and second part of Eq. (1) for a linear chirp of , one can get the followed expression by introducing an auxiliary function P = U + iV.
When Eq. (2) is Fourier integrated from -∞ to + ∞ by , one can get,
By solving the Eq. (3), the fast-chirped readout signal in frequency domain can be expressed as shown,
One can see that Eq. (4) explains the distorted signal due to a fast-chirped readout perfectly. The expression is same with the results by a perturbation theory . This means our model can work well for the analysis of the chirped readout signal.
Hence, a recovery process can be developed from Eq. (4) for the recovery of original input signal from the distorted readout signal, in which, firstly, K(ω) is obtained by Fourier transformation of the readout signal from the detector, secondly, the inverse Fourier transform of the multiplication of K(ω) by is carried out, finally, one can get the undistorted input signal. In the recovery model, the real part of K(ω) and H(ω) are utilized.
3. Experiment and discussion
An experiment setup shown in Fig. 1 is designed to validate our recovery model based on Bloch Equations. In which, a SHB crystal of Tm3+:YAG [7–9,14] is mounted in a He flow cryostat at a temperature of about 4.0 K. A stabilized external cavity diode laser (ECDL) is used as the write laser. The optical signal from the write laser is injected into an EOM by a fiber. The RF signal from the arbitrary wave-form generator (AWG) is modulated onto the optical signal by EOM. Next, the optical signal with RF pass through a FC and focus into the SHB crystal by L1. Another frequency-chirped ECDL is used as the read laser (Toptica DLC DL pro 780 nm with a mode-hop free tuning 30GHz). The optic from the read laser is splitted into a read beam and a reference beam by a PBS. The read beam probes the spectral hole burning which produced by the write beam, and the reference beam probed the unchanged inhomogeneous profile of the crystal, and then the two beams are received by a differential detector with a bandwidth of 80 MHz. The subsequent digitizing oscilloscope records the detected intensity, which is a temporal map of the signal power spectrum. Two acousto-optic modulators (AOM) work as optical switches to implement the write process and the read process, and a synchronization controller controls the open and close time of AOM1 and AOM2.
If a single signal is injected into the experimental setup, the experimental results are as shown in the blue solid lines of Fig. 2. One can observe the signal distortion for different readout chirp rates of 1.11MHz/us, 3.17MHz/us and 5MHz/us. The distortion shows more severely with the chirp rate increase.
Our recovery method is implemented to recover the distortion of the signals for different sweep rate, which shows a perfect recovery for the distortions as shown the red solid line of Fig. 2.
In order to validate our recovery model more deeply, a two tone signal at 200 MHz and 201 MHz with the same power of −10 dBm. The results are plotted in blue line of Fig. 3. It can be seen, the two spectral features are difficulty to distinguish using a chirp rate κ = 3.17 MHz/μs to readout. The recovered data is plotted in red line of Fig. 3 using our model, one can identify the two spectral features clearly. The case of a two-tone signal with an identical power demonstrates the validity of the recovery algorithm.
In conclusion, a model based on the Bloch Equations is presented for the case of fast-chirped readout in a spectrum analyzer using spectral-hole-burning, which is used to analyze and explain the distorted signal due to a fast-chirped readout. A recovery algorithm is developed using the model on Bloch Equations. The experiments with a single and two tone signal are performed to validate the recovery algorithm.
National Key Scientific Instrument and Equipment Development Projects of China (2014YQ090709); National Natural Science Foundation of China (11574328); and Beijing Jiaotong University Basic Scientific Research Foundation (2016RC046).
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