Reaching the Frequency Resolution Limit in a Single-Shot Spectrum of an Ultra-Short Signal Pulse Using an Analog Optical Auto-Correlation Technique

This work demonstrated a high-resolution spectrum obtained from a single ultra-short high-frequency signal pulse using a new analog optical auto-correlation method that revolutionizes the signal detection and measurement technique. It provides unprecedented capability to reach the fundamental limit in Physics. This technique allows us to increase the observation/measurement time by converting a Radio Frequency (RF) signal in the optical domain and keeping the “live-signal” in a fiber-optic-recirculation-loop circuit over an extended time that could be a million times longer than the pulse width. The dispersion delay of the two RF modulation sidebands is used to perform an analog time correlation. It effectively stretches time by million times. As a result, it provides a frequency-space “snapshot” of ultra-short events without the need for a high-speed analog-to-digital converter (ADC). A single 30 ns-short two-tone pulse separated by 25 MHz was resolved in a 50 GHz wideband real-time frequency spectrum. This technique allowed us to push the physical limit in the spectral resolution of an ultra-short signal and enable a new technology to manipulate, process, and analyze ultra-short electromagnetic signals and transient events. It also provokes a fundamental question: Is the uncertainty principle limiting the frequency determination for a wave in a continuum state?


I. INTRODUCTION
C ONVENTIONAL electronic receiving and processing techniques detect an electromagnetic wave signal by converting the live wave signal to a digitized current (voltage) signal using an ADC and performing signal processing in the digital domain. If the signal's frequency is higher than the ADC's sampling rate, repeated signal sampling must be employed to construct the frequency spectrum. When a signal is only available for a short duration, Δt, such as a single ultra-short pulse, the conventional system's maximum signal detection time is limited by Δt, and the speed of the ADC limits its signal processing capability. As a result, it cannot provide a high resolution (< 1/Δt) real-time spectrum for an ultra-short signal that the frequency is higher than the sampling rate. Several photonic techniques have been developed to perform the spectrum analyzation or analog-to-digital conversion in the optical domain without using high-speed electronic ADC to obtain a real-time high-frequency spectrum. [1], [2], [3], [4], [5], [6], [7] However, these photonic techniques use frequency-domain signal processing or filtering in which the signal's pulse width still limits the signal detection time. Therefore, their frequency resolution will be limited by the ultrashort duration of the signal. We have introduced a new analog time-domain signal auto-correlation processing technique with a real-time measurement system " following" the single live electromagnetic wave pulse. [8], [9], [10], [11], [12] This can be realized by carrying live electromagnetic pulses in the optical domain using a fiber-optical recirculation loop circuit to perform analog correlation. In this case, the observation/measurement time is not limited by the pulse width. One can increase the frequency resolution of the measurement system by increasing the "observation" time and, therefore, provide a high-resolution, wideband frequency spectrum from a single short RF signal pulse without a high-speed ADC. Fig. 1 illustrates the conceptual analog signal auto-correlation processing technique using the optical-fiber recirculation loop circuit. A single short RF pulse signal modulates a laser carrier signal and generates a pair of input signals by two RF sidebands in the optical domain. The signal pairs are sent into a fiber optic recirculation loop and travel at slightly different speeds due to the dispersion of the optical fiber. After making a large number of circulations and tapping a small portion of the signal pulse replica for each round-trip, a photodetector converts the optical signal pulses into a series of RF signal pairs. The signal pairs with relative dispersion delays are sent into a square-law RF detector to make multiplication and integration. Therefore, it performs This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ a time-domain signal auto-correlation over an extended period. [8], [9], [10], [11], [12] The analog correlation signal is the base-band output signal from the square-law detector, which a low-speed ADC can digitize. A fast Fourier transformation (FFT) on the base-band correlation data is performed to obtain the frequency spectrum of the original signal. [12] Since the dispersion time delay between the sidebands is about six orders of magnitude smaller than the light's travel time in the fiber-loop, this technique "stretches" the time by a million times. This allows us to analyze a high-frequency signal (∼50 GHz) using a low-speed (50 MHz) ADC.
To obtain high-resolution spectra, thousands of circulations are needed to produce enough pulse replicas as the correlation terms. However, the signal taping and the loss in the loop circuit cause a rapid exponential decay in signal strength for the replicas. An Er-doped fiber amplifier (EDFA) can be used in the loop circuit to compensate for the loss and maintain the optical pulse intensity. Unfortunately, the EDFA introduces optical noise into the loop system, which will be amplified in subsequent loop circulations. Consequently, even if the loop circuit can produce thousands of optical pulse replicas, the signal-to-noise ratio degrades so rapidly that the signal is no longer useful for correlation processing. Therefore, increasing the number of replicas with a valid RF signal-to-noise ratio becomes a significant challenge for this technique.
To solve this problem, this work used an improved RF-photonic fiber-recirculation-loop autocorrelation circuit architecture to improve the signal-to-noise level. This new architecture stores and reinject the single input signal in the time domain, allowing the signal-to-noise ratio to be restored periodically, producing thousands of correlation terms. The detailed description of this new architecture and its advantages is disclosed in a U.S. patent [13]. This architecture demonstrated a high-resolution spectrum for a single short signal pulse. In addition, a new noise reduction method was employed by adding both time domain and frequency domain noise filters to reduce the signal's amplitude fluctuation caused by the EDFA noise and improve the stability of the system. Moreover, the experiments were conducted in the temperature-controlled laboratory on a vibration isolated optical table to avoid any environment induce noise. These measures significantly increased the number of replicas, therefore increasing the resolution.
The main objective of this work is to explore the limit of the spectrum resolution from a single ultra-short signal pulse. However, when the signal is ultra-short, it limits the overlapping time to perform the time auto-correlation, reducing the resolution. This work introduced a delayed copy method to artificially expand the signal's pulse width, increasing the overlapping time of the auto-correlation process. This will not affect the frequency information after the Fourier transformation. These new methods allowed this optical analog auto-correction-based spectrum analyzing system, for the first time, to obtain a high resolution (< 1/Δt) real-time spectrum for several ultra-short (∼30 ns) signals that the frequencies (≥ 30 GHz) are higher than the sampling rate. The results also provoke a more fundamental question: according to the Heisenberg-Gabor limit, [14] the frequency resolution cannot be better than 1/(4πΔt). If we use this technique to increase Δt by an artificially extended observation/measurement time, would it bypass the limit set by the uncertainty principle?

II. CONCEPT AND EXPERIMENTAL SYSTEM ARCHITECTURE
The concept of the reinjecting recirculation-loop circuit is shown in Fig. 2. A CW laser is modulated by an RF signal pulse, E(ω RF t), via a LiNbO3 Mach-Zehnder modulator so that a "live" RF signal is carried by a laser beam which is gated by a fast 1 × 2 switch to create a "square" optical pulse. The input optical pulse enters a 2 × 2 coupler such that a small portion, ∼5%, is sent into an unamplified recirculation fiber loop; the rest of the signal bypasses the loop and enters into a long cueing delay fiber loop via a second 1 × 2 switch at the first loop's output. The unamplified recirculation loop is made by a ∼1.34 km long fiber and reconnects to the second input of the 2 × 2 coupler. The loop output is sent to the second 1 × 2 switch that directs a string of 24 pulse replicas to the second output of the 1 × 2 switch that connects to a photodetector, followed by a step amplifier. The loop loss and signal tapping cause an exponential decaying of the 24 pulses' amplitude, eventually decaying to the system's noise floor level. The remaining large portion of the input pulse travels in a long cueing fiber loop, with a delay time of 24 times the recirculation loop's round-trip time. An EDFA then amplifies the pulsed signal before reinjection into the second input of the 2 × 2 coupler via the 1 × 2 gate switch. Another 24 pulse replicas are then produced, following the first 24 replicas. The reinjected pulse generates the following 24 replicas that maintain the same time interval between subsequent replica pulses. This process is repeated for every 24 recirculation-loop trips so that the replica signal-to-noise ratio increases periodically instead of only decaying exponentially. [9], [13] The two RF modulation sidebands are separated from the optical carrier by the RF frequency, f RF . The two RF sidebands travel through the fiber at different speeds due to dispersion, which causes a relative time delay, "δt," for each loop round trip. The dispersion relation is quite linear in a regular optical fiber (Corning SMF-28) at 1.55μm wavelength. Therefore, we can express it as δt = f RF τ = f RF Lτ o , where L is the length of the fiber, τ = Lτ o, and τ o is a constant; at 1550 nm, τ o ≈ 28.85 × 10 −26 s 2 /m. [9], [15] This loop circuit creates a train of pulses with RF sidebands pairs having δt, 2δt, 3δt . . . nδt, delays between the sidebands, where n is the number of pulse replicas generated. The RF signal carried by both sidebands is extracted from the optical carrier with a photodetector. For each replicated pulse, these outputs are the correlation coefficient between a pair of RF signals e iω RF t and e i[ω RF (t+kδt)] , where k represents the k th replica and ω RF = 2πf RF . After amplification, the signal is sent to an RF square-law detector to perform the multiplication: With a low pass filter, the baseband signal contains the correlation product, 1 + cos(ω RF kδt), which can then be expressed as 1 + cos[(ω RF ) 2 kτ ]. Therefore, one can perform a simple Fourier transformation between time "kτ " and frequency squared (ω RF ) 2 to obtain the frequency spectrum for the input RF signal.
The periodic change to every group of 24 pulses caused by the long-loop, amplified signal injection to the un-amplified loop circulation will produce a series of "spurious" frequencies after the Fourier transform. We must correct the artificial exponential decay behavior by a normalization process to eliminate these spurs. In principle, we know the optical loss for each loop transit and the period, so we should be able to normalize the data in the digital domain. This work introduces a step-gain amplifier on the base-band correlation output signal to perform an analog normalization process. It significantly reduced the spurs. However, we need to normalize the RF signal mixed with an unknown noise floor dynamically fluctuated by the EDFA in the optical domain. Noticeable spurious peaks will still appear in the RF spectrum after the FFT.
A new noise reduction method was employed by adding both time domain and frequency domain noise filters to improve signal-to-noise performance. These measures significantly increased the number of replicas, therefore increasing the resolution.

III. EXPERIMENT RESULTS
A series of experiments were conducted in which two-tone RF pulses are generated by two RF signal generators (Agilent E8257N) that are synchronized by an external trigger signal. The same trigger is also used to control the recirculation-loop circuit switches. Two RF signal tones of frequencies are combined with an RF combiner to create a single RF pulse containing two RF tones that are used to modulate the optical signal of the recirculation-loop-circuit system that performs the analog signal autocorrelation and FFT. During these experimental tests, several center frequencies from 26 GHz to 40 GHz and various two-tone frequency separations from 25 MHz to 100 MHz were used for the combined two-tone RF pulse. The goal was to test the resolution limit of our system with an ultra-short single pulse. To increase the resolution, we need to increase the number of recirculation to generate more pulse replicas. The improved experimental set-up, shown in Fig. 2, can generate 4096 replica pulses.
The resolution of these spectra is not a constant since we did a Fourier Transformation from time-space to frequency squared space. With 4096 data points in the time domain data and ∼50 GHz of total bandwidth defined by the dispersion time difference between the two sidebands in one loop roundtrip, we can obtain a spectrum resolution of 20 MHz or better for frequencies above 18 GHz. This theoretical resolution has also been verified experimentally.
When the RF signal pulse width is reduced to 30 ns, at 30 GHz frequency, the time delay between the two RF sidebands after one loop roundtrip is Δt = 0.0116 ns. Therefore, after 3000 loop trips, the two RF sidebands will be separated by more than 30 ns, and they can no longer overlap in time to be mixed by the square-law detector. An interleaving scheme is used to increase the overlapping time, as shown in Fig. 3. The 30 ns short two-tone RF pulse is sent to split the signal into a -15 dB 1 x 2 directional coupler. The primary output signal is sent into a ∼6.5 m long RF delay-line cable (having ∼15 dB loss), combined with the -15dB output of the coupler using a 2 x 1 RF combiner followed by an RF amplifier before connecting to the optical modulator. The combined signal interleaves the original 30 ns pulse with a ∼30 ns delayed copy. Therefore, the new RF pulse width becomes 60 ns. Note that the delay-patched pulse may have an added phase. During the correlation process, that phase generates a constant term that will not change the RF signal's frequency position after FFT in the frequency spectrum; it only increases the intensity of the DC signal peak at the origin of the RF spectrum. Fig. 4 shows the experimental time auto-correlation data of a 30 ns short RF input signal pulse containing two tones of 29.975 GHz and 30.000 GHz; the output correlation pulse signals are collected at the output of the step amplifier with a real-time oscilloscope. The first few hundred correlation pulses out of 4096 total pulses are shown in Fig. 4. The insert shows the zoom-in detail of those pulses, which is the correlation of the ultra-short RF pulse. The residual noise caused by the wider optical pulses is the small footing under each pulse. The time axis is represented by sampling points with a sampling rate of 250 MS/s. The y-axis is amplitude with an arbitrary unit. The pulses' envelop shape shows the frequency square's cosine behavior. Note that a 250 MS/s sampling rate allows us to measure a single peak position with only four ns accuracy. To reduce the stitching error between the long buffer delay loop and the recirculation loop's 24 round trip travel, experimental data of the 4096 peaks were used to fit the calculated peak positions by fitting the recirculation loop's round trip travel time and stitching error for every 24 loops travel. This allows us to determine the stitching error down to ps level accuracy. Therefore, we can experimentally adjust the long buffer delay Fig. 4. Experimental data of the auto-correlation output was captured from a single shot real-time oscilloscope using a 250 MHz data rate. The insert on top shows a "zoom-in" detail of the first ∼50 pulses. The red dots indicate the peak intensity; the green dot indicates the correspondent noise floor.
fiber's length to mm precision to reduce the stitching error. After the adjustment, the calculated peak positions of 4096 pulses showing as red dots matched the experimental data's peaks.
The raw data curve produces 4096 time-correlation data points by taking each peak intensity and subtracting the noise footing value. A Fourier transformation can be performed on the 4096 time-correlation data points to obtain a 4096-point frequency square spectrum. The total frequency bandwidth can be calculated using the dispersion time between the two RF sidebands for the 1.34 km round trip travel in the loop, which give the total bandwidth of ∼ 54.8 GHz. We can convert the x-axis of the FFT data from frequency-square to frequency. Fig. 5 shows the frequency spectrum obtained from the FFT. Since the signal reinjection still makes residues of artificial periodic intensity changes for every 24 circulations in the unamplified loop system, a spur frequency of ∼15.4 GHz is generated in the frequency spectrum (15.4 GHz) 2 in the spectrum after direct FFT. It also generated the harmonics of N x (15.4 GHz) 2 spurs in the frequency-square spectrum. These spurs can be referred to as system spurs. Those spurs are shown in the square root of N x 15.4 GHz in the frequency spectrum. Therefore, those spurs can be easily identified. The ∼30 GHz signal peak is clearly shown in the spectrum. Even after a step-amplifier is used to reduce the periodic artificial changes, the first three spur peaks are higher than the signal peaks. Because the input signals are very short (∼30 ns), the amplitude of the RF signal peak is weak. In addition, the signal is mixed with the system spurs to generate additional spurs. Fig. 6 shows the "zoom-in" high-resolution RF spectrum around 30 GHz. The 29.975 GHz and 30.000 GHz two-tune 30 ns short pulses are clearly shown in the spectrum. There are few spurious peaks caused by the two tones shown on the noise floor.
The spectrum showed better than 25 MHz resolution for a single 30 ns short pulse two-tone high frequency (∼30 GHz)   input signal. Therefore, a better than 1/ Δt frequency resolution is resolved without using 30 GHz ADC.
This experimental result is a representative data of many experiments conducted with different RF-signal frequencies, different two-tone separation, or pulse width. Fig. 7 shows another experimental result of zoomed-in FFT spectrum from a 30ns two-tone pulse centered at 39 GHz with 100 MHz tone separation. At higher frequency, the peaks' amplitude was lower than those from lower frequency data due to the RF loss from the components used in the system. However, two peaks separated by 100 MHz can be clear observed above the noise level.
To analyze the errors in our measurement precision, we have considered many factors that can contribute to the errors in the experimental system. From our extensive previous experiments, the major factor is the EDFA noise; the secondary factor is related to environment perturbation that affect the fiber optic system. The mitigations these noises have been already discussed in this paper. The next order of noise to be considered is the added phase noise from the RF-Photonic link. This includes laser noise (linewidth broadening), noise caused by modulator, fiber, detector, and RF amplifiers. These have been carefully studied in our previous work in reference [16], which indicates that the level of noise added is negligible. Since the noise in the experiment depends on the input signal's frequency and amplitude, a practical method can be used to estimate the error bar. From Fig. 7, we can see two peaks of the two RF tones. From the data, we can calculate the average linewidth of those peaks which is about 10 MHz. This should be the upper limit of the error bar in this experimental result.

IV. DISCUSSION
So far there is no commercial RF spectrum analyzer that can provide 50 GHz wideband spectrum from a single 30ns short pulse with 10 MHz accuracy, because traditional spectrum analyzer must work with a narrow band system and scan band by band or perform multiple under samplings to add them into a wideband spectrum, it does not have enough time to deal with a very short signal pulse. Our system can make a single-shot 50 GHz real-time spectrum with 10 MHz linewidth for a short signal. The matter is that our system allows us to determine the absolute frequency of the signal with a Δf/f of <10 −6 accuracy. To resolve two-tones, 25 MHz resolution with 10 MHz error-bar is still meaningful in Physics when investigating short event or transient signal.
People generally believe that the uncertainty principle applies to all waves because the wavefunction in the two corresponding orthonormal based in Hilbert space are Fourier transforms of one other, which cannot both be sharply localized. Therefore, the frequency resolution is set by the Heisenberg-Gabor limit, 1/(4πΔt). [14] However, this paper demonstrated that we could expend the localized time interval for waves in a continuum state without changing the wave's energy state (frequency). This experiment indicates that the recirculation loop-based correlator can generate an effective signal "duration" Δt. The signal can be stretched in time much larger than the pulse width Δt. Therefore, increasing the number of circulations in the loop and making more pulse interleave using this recirculation-loop-based analog auto-correlation technique could resolve the Heisenberg-Gabor limit.
The uncertainty principle indicates that for a physical quantity pair such as position-momentum or time-frequency (energy), they cannot become arbitrarily small simultaneously to be determined. There is an accuracy limit to determining one quantity while measuring the second quantity with an arbitrarily small precision range in a quantum regime. That is because a particle in a bounded state has quantized discrete energy levels. However, in the case of a continuum, such as the propagating electromagnetic wave pulse (not in a bounded state) in our experiment, there is no sudden change in the energy state (frequency). Using our technique to "follow" a "live" signal pulse for an "observation" time much longer than the pulse width can result in better resolution in frequency determination than the traditional spectrum analyzing method.

V. CONCLUSION
This work demonstrated an optical-recirculation loop-based analog auto-correlation spectrum analyzation system that, for the first time, achieved a high-resolution spectrum that resolved 25 MHz two-tone for a 30 ns duration short high frequency (∼30 GHz) RF pulse. This optical analog time correlation technique will allow us to "stretch" time and avoid needing a high-frequency ADC to obtain a "snapshot" in the frequency domain for ultra-short or transient events. This also indicates that it is possible to "bypass" the limit set by the uncertainty principle in the case of a continuum by extending the "observation/measurement" time using this RF-Photonic analog autocorrelation technique. This technology opens the door for many new type of detection/measurement system applications that allow studying ultra-short transient events. It may revolutionize the future RF electronic systems.