Systems and methods for lock-in detection for spectra from diverse spectrometers across the electromagnetic spectrum

Abstract

Systems and methods for lock-in detection of an electromagnetic signal are described. In some embodiments, a method includes receiving a signal at a reference center frequency, decomposing the received signal into frequency component signals in time domain by a fast Fourier transform (FFT), a discrete Fourier transform (DFT), or a set of filters, demodulating each of the frequency component signals by multiplying it with the in-phase (I) and / or quadrature (Q) components of the reference signal, and passing each demodulated frequency component signal through a low-pass filter.

Description

Systems and Methods for Lock-in Detection for Spectra from Diverse Spectrometers Across the Electromagnetic SpectrumFIELD OF THE INVENTION

[0001] The present invention relates generally to signal processing of periodic electromagnetic signals and more specifically to lock-in detection to extract an electromagnetic signal obscured by noise.BACKGROUND OF THE INVENTION

[0002] Lock-in detection is a signal processing technique designed to extract weak signals from noisy environments by isolating the desired signal at a specific reference frequency. It involves mixing the input signal with a reference signal and then applying a low-pass filter to recover the amplitude and phase of the signal of interest. By rejecting noise outside the reference frequency band, this method significantly improves the signal-to-noise ratio (SNR), enabling the detection of signals that would otherwise be obscured by broadband noise. However, conventional lock-in amplifiers typically do not preserve spectral information because they isolate and extract signals within a very narrow frequency band centered on the reference frequency. While this makes them effective for measuring narrowband signals, they are less suitable for applications requiring broad spectral analysis, such as heterodyne terahertz spectrometry.SUMMARY OF THE INVENTION

[0003] Systems and methods for lock-in detection of an electromagnetic signal are disclosed. In many embodiments, the method includes receiving a signal at a reference center frequency, decomposing the received signal into frequency component signals in time domain by a fast Fourier transform (FFT), a discrete Fourier transform (DFT), or a set of filters, demodulating each of the frequency component signals by multiplying it with the in-phase (I) and quadrature (Q) components of the reference signal, and passing each demodulated frequency component signal through a low-pass filter.

[0004] In some embodiments, the method also includes dividing the modulated signal into time windows of duration Tseg, each containing N samples.

[0005] In some embodiments, the noise power of the lock-in detected frequency bins decreases at a rate of 10 dB per decade with increasing integration time while the signal power remains constant.

[0006] In some embodiments, the mean demodulated voltages or current at each frequency bin remain constant while the corresponding noise voltage is reduced at a rate of 5 dB per decade as a function of integration time.

[0007] In some embodiments, the frequency components of the signal remain constant over an integration time such that the signal bandwidth of each frequency bin approaches an infinitesimally small value.

[0008] In some embodiments, the method also includes calculating a magnitude of voltage at each frequency bin to account for phase incoherence in the signal.

[0009] In some embodiments, a low-pass filter bandwidth is larger than a signal bandwidth, such that increasing an integration time reduces noise power while maintaining signal power.

[0010] In some embodiments, the method also includes a signal modulation step at the reference frequency to bring the signal at the reference center frequency before demodulation.

[0011] In some embodiments, the method also includes a frequency shifting step to bring the signal to the reference center frequency before demodulation.

[0012] In some embodiments, the method also includes integrating each demodulated frequency component signal over time.

[0013] In some embodiments, the method also includes referencing the reference and received signals and demodulating by multiplying the received signal with I or Q components of the reference signal.

[0014] In some embodiments, the received signal is converted to the digital domain using an analog-to-digital converter and the signal processing is performed using an FPGA (field programmable gate array).

[0015] In some embodiments, the received signal is converted to the digital domain using an analog-to-digital converter and the signal processing is performed using a PC using different programs (e.g., Matlab, C++, Python).

[0016] In some embodiments, the input signal is received from a photonics receiver (e.g., photonic detector, a photomixer, or a photodiode).

[0017] In some embodiments, the photonic receiver includes a photomixer driven by a heterodyning optical beam.

[0018] In some embodiments, the reference signal is a square wave.

[0019] In some embodiments, the reference signal is a sinusoidal wave.

[0020] In some embodiments, decomposing the modulated signal into frequency component signals produces distinct frequency components fi, fz, ..., u-

[0021] In some embodiments, the method also includes calculating an amplitude R(f) and a phase cp(f) for each frequency component, where R(f) = ^(X(f)2+ Y(f)2) and cp(f) = atan[Y(f) / X(f)], and where X(f) and Y(f) represent outputs obtained by mixing each frequency component with the in-phase and quadrature reference signals, respectively.

[0022] In some embodiments, the low-pass fdter is implemented through signal integration, and a filter bandwidth is inversely proportional to an integration time.

[0023] In some embodiments, the optical receiver pumped by the heterodyning optical beam detects terahertz signal.

[0024] In some embodiments, the method also includes dividing the received signal into time slices smaller than switching intervals of the reference signal, where the time slices are based on a desired frequency resolution.BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Fig. 1 illustrates a diagram and operation of a conventional lock-in detection system.

[0026] Fig. 2 illustrates a hardware diagram of a spectral lock-in detection system in accordance with an embodiment of the invention.

[0027] Fig. 3 illustrates a diagram and operation of a spectral lock-in detection system in accordance with an embodiment of the invention, which demodulates the magnitude of the received voltage.

[0028] Fig. 4 is a flow chart illustrating a process for spectral lock-in detection in accordance with an embodiment of the invention.

[0029] Fig. 5 is a graph illustrating results of spectral lock-in detection on data captured by a plasmonic heterodyne spectrometer in accordance with an embodiment of the invention.DETAILED DISCLOSURE OF THE INVENTION

[0030] Lock-in detection techniques in accordance with embodiments of the invention are disclosed that can enhance the SNR of captured spectra while retaining spectral information. In further embodiments, methods are applied to data from a plasmonic heterodyne terahertz spectrometer to demonstrate a noise reduction rate of 10 dB per decade of integration time, with the spectral integrity fully preserved.

[0031] As mentioned further above, a conventional lock-in detection system aims to reduce SNR by moving the center frequency of the signal to a range that has less background noise. A block diagram and operational principle of a conventional lock-in detection system is illustrated in Fig. 1. On the left side, the spectrum of a signal with background noise is depicted. By modulating the signal at a frequency fnod, the center frequency of the signal shifts by the modulation frequency. Selecting an appropriate modulation frequency allows the signal to move away from flicker noise and other low-frequency noise components, thereby significantly enhancing the SNR. The block diagram on the right side in Fig. 1 demonstrates how the modulated signal, Vmoduiated(t), is demodulated using a reference modulation signal, Vrej t), to extract the amplitude R and phase q> of the original signal with improved SNR. A filtering stage averages the signal over time.

[0032] Typically, the demodulation in these conventional systems must use the same frequency that was used for modulation fmod. Accordingly, the system must be coherent, which can be difficult in keeping the receiver informed of the modulation frequency of the signal.

[0033] In conventional lock-in detection, the relationship between the low-pass filter bandwidth and the signal bandwidth affects the SNR improvement rate. At short integration times,when the low-pass filter bandwidth is larger than the signal bandwidth, increasing the integration time reduces the noise power at a rate of 10 dB per decade while maintaining the signal power, resulting in an SNR improvement of 10 dB per decade. However, at longer integration times, when the low-pass filter bandwidth becomes smaller than the signal bandwidth, both the noise power and the signal power decrease at a rate of 10 dB per decade, preventing any further enhancement in SNR with increasing integration time. In either regime, the lock-in detected output represents a superposition of all filtered frequency components and does not preserve the spectral information of the original signal.Lock-in Detection with Spectral Information

[0034] A block diagram illustrating operational principles of a spectral lock-in detection system in accordance with an embodiment of the invention is illustrated in Fig. 2. On the left side, a time-domain signal showing signal power versus time is displayed, featuring a modulated waveform with varying amplitude levels. This time-domain signal undergoes Fast Fourier Transform processing. The middle section shows the result of the FFT decomposition, displaying multiple separate graphs representing the power of individual frequency components fz, f?, through fv, each plotted against time. These frequency-separated signals maintain the same modulated pattern as the original input signal. The right section illustrates the demodulation process for each frequency component, showing parallel processing channels. Each channel contains a reference signal Ere / / ) that is split into two paths, one direct and one phase-shifted by 90 degrees, which are then multiplied with the frequency component signal using mixers. The outputs of these multiplications feed into low-pass filters, producing X and Y outputs for each frequency. These X and Y values are then used to calculate the amplitude R(f) and phase cp(f) for each respective frequency component.

[0035] Fig. 3 presents a visualization of the spectral lock-in detection workflow. At the top left, a raw time-domain modulated signal v(t) is shown as a series of sampled voltage values over time. This signal is segmented into multiple time windows, each containing N samples. For each time window indexed by m, a discrete Fourier transform (DFT) is applied to extract the frequency components, producing voltage values at different frequency bins denoted as Vzfm). The diagramillustrates extracting the frequency components of each time window, compiling the time-domain variation of each frequency component across all windows, and applying lock-in detection to each frequency bin's time sequence. The demodulation process involves multiplication with in-phase (I) and quadrature (Q) components of the reference modulation signal, represented by sgn functions of sine and cosine terms at the modulation frequency. The visualization demonstrates how the spectral information is preserved while enabling lock-in detection to be applied independently to each frequency bin's time sequence.

[0036] A process for lock-in detection to resolve captured spectra with enhanced SNR in accordance with an embodiment of the invention is illustrated in Fig. 4. The process 400 includes recording (402) the raw time-domain modulated signal, which is produced by modulating the original signal. The spectral lock-in detection process may begin by dividing the sampled voltage into time windows of duration Tseg, each containing N samples. For each time window, the frequency components are extracted using a discrete Fourier transform (DFT) or a Fast Fourier Transform (FFT), which decomposes (404) the time-domain signal into distinct frequency components, / , fi,AS can be seen in the graphs, a frequency component may exhibit on- off cycles. The signal should be divided into time slices that are smaller than the switching intervals and are based on the desired frequency resolution. For example an 8 microsecond time slice leads to a 125 kHz frequency resolution. Increasing the time slice would give a higher frequency resolution (narrowed bandwidth for each frequency bin).

[0037] Each time series of a frequency component is demodulated (406) by multiplying it with the in-phase (I) and quadrature (Q) components of the reference modulation signal, Vreflf). After extracting the frequency components for each time window, the magnitude of the voltage at each frequency bin may be calculated to account for possible phase incoherence in the signal. This process obtains the original amplitude, R(f) = ^ / 7)2+ F( )2and phase, <p( ) = atan[Y (f) / X ( / )], with improved SNR. Here, X(f) and Y(f) represent the outputs obtained by mixing each frequency component with the I and Q reference signals, respectively, followed by low-pass filtering (408) to eliminate out-of-band noise. This low-pass filtering can be implemented digitally or through signal integration, where the filter bandwidth is inversely proportional to the integration time.

[0038] Because lock-in detection is applied to the time stream of each individual frequency bin, the noise power of the lock-in detected bins may decrease at a rate of 10 dB per decade with increasing integration time, while the signal power remains constant. This occurs because, assuming the frequency components of the target signal remain constant over the integration time, the signal bandwidth of each frequency bin approaches an infinitesimally small value (i.e., a delta function). Consequently, the demodulated signal power at each frequency bin remains constant, independent of the integration time. The mean demodulated voltages or currents at each frequency bin remain constant, while the corresponding noise voltage or noise current is reduced as a function of integration time at a rate of 5 dB per decade. Similarly, the mean power corresponding to the demodulated voltages or current at each frequency bin remains constant, while the corresponding noise power is reduced as a function of integration time at a rate of 10 dB per decade. This leads to a corresponding SNR improvement of 10 dB per decade while preserving frequency-domain information.

[0039] Although a specific process is described above with respect to Fig. 4, one skilled in the art will recognize that any of a variety of processes may be utilized for spectral lock-in detection in accordance with embodiments of the invention.

[0040] Results of spectral lock-in detection on data captured by a plasmonic heterodyne spectrometer in accordance with an embodiment of the invention are illustrated in Fig. 5. The photomixer is driven by a 75 mW heterodyning optical beam at -785 nm wavelength, with a beat frequency near 680 GHz. The terahertz input is generated using an amplifier / multiplier chain, producing a tone at 678.796 GHz. Eccosorb sheets and a Mylar film can be used to attenuate the terahertz signal to -120 dBm. The terahertz tone is modulated at 125 Hz by modulating the RF input of the AMC using a 125 Hz square wave from a function generator. The intermediate frequency (IF) output from the plasmonic heterodyne spectrometer is processed using a spectral lock-in technique as discussed further above.

[0041] Fig. 5a shows the resolved IF spectra with the beat frequency at 0, with a spectral resolution of 125 kHz, at various integration times. At low integration times, the IF signal is dominated by broadband noise, primarily from the Johnson-Nyquist noise of the photomixer and laser noise, causing the IF tone to be buried under this noise. However, as the integration timeincreases, the noise level decreases at a rate of 10 dB per decade, revealing the IF tone after just 1 second of integration. Further increasing the integration time beyond 1 second continues to reduce background noise and improves the SNR of the resolved IF spectrum. As illustrated in Fig. 5c, the SNR of the lock-in detected IF tone increases from 8 dB to 24 dB when the integration time is increased from 1 second to 320 seconds. Figure 5b shows the resolved IF spectrum after 320 seconds of integration. This is particularly notable given the room-temperature operation of the spectrometer, which corresponds to an input noise power density of -122.8 dBm at a spectral resolution of 125 kHz.

[0042] The double-sideband (DSB) noise temperature of the spectrometer is calculated from the SNR of the resolved IF spectra at different integration times, while accounting for the input power of -120 dBm. The DSB noise temperature decreases from 91 K to 2.5 K when the integration time is increased from 1 second to 320 seconds, surpassing the performance of cryogenically cooled heterodyne terahertz spectrometers. The spectral lock-in detection methods described herein in accordance with embodiments of the invention are versatile and can enhance the SNR of plasmonic heterodyne terahertz spectrometers across various frequency bands. They offers quantum-level sensitivity crucial for applications in astronomy, cosmology, and atmospheric studies, without the need for cryogenic cooling — a limitation that currently restricts the scope and potential usage of heterodyne terahertz spectrometers. The presented spectral lock-in detection techniques can also enhance the SNR of other spectrometers operating at other parts of the electromagnetic spectrum, with a wide range of applications in sensing and hyperspectral imaging.

[0043] Although the description above contains many specificities, these should not be construed as limiting the scope of the invention but as merely providing illustrations of some of the presently preferred embodiments of the invention. Various other embodiments are possible within its scope. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents.

Claims

WHAT IS CLAIMED IS:

1. A method for lock-in detection of an electromagnetic signal, the method comprising: receiving a signal at a reference center frequency;decomposing the received signal into frequency component signals in time domain by a fast Fourier transform (FFT), a discrete Fourier transform (DFT), or a set of filters;demodulating each of the frequency component signals by multiplying it with the in-phase (I) and / or quadrature (Q) components of the reference signal; andpassing each demodulated frequency component signal through a low-pass filter.

2. The method of claim 1, further comprising dividing the received signal into time windows of duration Tseg, each containing N samples.

3. The method of claim 1, where the noise power of the lock-in detected frequency bins decreases at a rate of 10 dB per decade with increasing integration time while the signal power remains constant.

4. The method of claim 1, where demodulation is applied to the received voltage or current and the mean demodulated voltages or currents at each frequency bin remain constant while the corresponding noise voltage or noise current is reduced at a rate of 5 dB per decade as a function of integration time, resulting in a noise power that is reduced at a rate of 10 dB per decade as a function of integration time.

5. The method of claim 1 , where demodulation is applied to the received power and the mean demodulated powers at each frequency bin remain constant while the corresponding noise power is reduced at a rate of 5 dB per decade as a function of integration time.

6. The method of claim 1, further comprising calculating a magnitude of voltage or current at each frequency bin to account for phase incoherence in the signal.

8. The method of claim 1, comprising a signal modulation step at the reference frequency to bring the signal at the reference center frequency before demodulation.

9. The method of claim 1, comprising a frequency shifting step to bring the signal to the reference center frequency before demodulation.

10. The method of claim 1, further comprising integrating each demodulated frequency component signal over time.

11. The method of claim 1, comprising demodulation by multiplying the received signal with I or Q components of the reference signal.

12. The method of claim 1, where the received signal is converted to the digital domain using an analog-to-digital converter and the signal processing is performed using an FPGA (field programmable gate array).

13. The method of claim 1, where the received signal is converted to the digital domain using an analog-to-digital converter and the signal processing is performed using a PC using different programs (e.g., Matlab, C++, Python).

14. The method of claim 1, where the input signal is received from a photonics receiver (e.g., photonic detector, a photomixer, or a photodiode).

15. The method of claim 14, where the photonic receiver includes a photomixer driven by a heterodyning optical beam.

16. The method of claim 1, where the reference signal is a square wave.

17. The method of claim 1, where the reference signal is a sinusoidal wave.

18. The method of claim 1, where decomposing the received signal into frequency component signals produces distinct frequency components f / , f?, ... , fv.

19. The method of claim 1, further comprising calculating an amplitude R(f) and a phase cp(f) for eachfrequency component, where R(f) = (X(f)2+ Y(f)2) and (p(f) = atan[Y(f) / X(f)], and where X(f) and Y(f) represent outputs obtained by mixing each frequency component with the in-phase and quadrature reference signals, respectively.

20. The method of claim 1, where the low-pass filter is implemented through signal integration, and where a filter bandwidth is inversely proportional to an integration time.

21. The method of claim 1, where the optical receiver pumped by the heterodyning optical beam detects terahertz signal.

22. The method of claim 1, further comprising dividing the received signal into time slices smaller than switching intervals of the reference signal, where the time slices are based on a desired frequency resolution.

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