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The paper describes a modification of a high-resolution heterodyne NIR spectrometer described by Rodin, et al. in Opt. Express 22, 13825 (2014), wherein the noise amplitude of the heterodyne signal squared was proportional to the power of the incoherent radiation source (the sun). The addition of a second receiver collecting radiation from the sun into a second single-mode fiber created up to a factor of two increase in detected source power spectrum. The ability to add uncorrelated heterodyne IF signals as Gaussian noise (variance) provides the means by which multiple heterodyne receivers of signal from an incoherent source can increase the detected source power by the same multiple, thus avoiding the limit imposed by the antenna theorem for a single receiver (A. E. Siegman, Appl. Opt. 5, 1588 (1966)).
© 2016 Optical Society of America
1. Introduction
Heterodyne spectroscopy using the sun as a broadband light source and a narrowband local oscillator has proved to be a valuable measurement technique for remote atmospheric sounding measurements. Beginning with the initial published work performed with a fixed-frequency CO2 laser [1], the technique has been demonstrated using many different local oscillator sources in the infrared (IR) spectral region where the fundamental absorption bands of many important atmospheric species are located. Studies of IR heterodyne spectroscopy have been published using lead-salt diode lasers [2] and, more recently, quantum-cascade laser sources [3]. Review papers [4,5] show the development of the technique over time for IR sources.
Heterodyne spectroscopy uses the mixing of light from a broadband source of radiation, such as the sun or moon, with narrowband radiation from a laser source. The beat frequency between these two light sources is detected on a photodetector and integrated in time. The signal received by the photodetector is a mixture of a DC component and a radiofrequency mixing component (traditionally called the intermediate frequency, IF) of the two signals. The IF beat signal is isolated from the DC, and any reduction in signal due to absorption in the atmosphere manifests as changes in the IF signal amplitude. The technique takes advantage of the narrow linewidth of the laser source to make absorption or emission measurements with much higher spectral resolution than can be achieved using a standard grating-based spectrometer. It also can, in principle, be shot-noise limited in that the laser shot noise can easily dominate over detector noise even at high spectral resolution.
In spite of the strength of the absorption lines in the mid-IR, there is a strong case for the development of heterodyne spectroscopy in the near-infrared region between 1200 and 1700 nm, particularly when a small, low-powered and robust sensor is required. In particular, we are interested in developing a sensor that is small enough to be deployed on a satellite for atmospheric limb-sounding measurements. In the near-IR range, advances in telecommunications technology has made available the stable narrowband distributed feedback laser sources, high-frequency, uncooled InGaAs photodetectors and single-mode optical fibers, couplers and attenuators that make this a desirable wavelength region for satellite-based heterodyne absorption experiments. Changing the laser wavelength to tune the detection system allows measurements to be obtained using a pair of photodetectors rather than the large CCD arrays required of a grating-based spectrometer.
Recently, near-infrared heterodyne systems have been successfully demonstrated by Rodin et al. [6] and Wilson et al. [7] for the investigation of 1651-nm CH4 and 1573.6-nm CO2 transitions respectively, to provide column density measurements for these gases. This paper follows the method described by Rodin et al., but with the addition of another fiber to couple in more sunlight to the system, increasing the signal-to-noise ratio of the measurement.
Repeating Eqs. (1) and (2) of [6], let D be detector responsivity, while ES and ELO are components of the electric field associated with the radiation of the observed source and the local oscillator (LO), respectively. Then the heterodyne IF signal component of the photocurrent may be expressed as the convolution of the source and LO radiation fields in spectral space:
where gLO and Fs are power spectral densities of the LO and signal, and φLO and φs are the respective phases. Due to random phase and intensity variations in thermal broadband radiation at any given emission frequency, ω, the heterodyne IF component of the photodetector current is an additional source of Gaussian noise on the signal over and above any inherent noise contributions produced by the electronics and by the intensity + shot noise of the LO source.Following the derivation of Siegmann [8], the heterodyne IF component squared is proportional to the product of the two mean (time-averaged) DC photocurrents that would be measured individually from each source. The result is equivalent to Rodin’s result:
where iLO represents the mean photocurrent that would be produced by the local oscillator, and B(di/dω) represents the mean photocurrent that would be produced by the thermal source. Here dis/dω is the mean photocurrent per unit frequency of the thermal source. The term B(di/dω) is proportional to the mean spectral power density in the bandpass, B, over which Fs is sufficiently constant.In the actual measurement performed here, neither of the DC photocurrents is measured. Because the heterodyne IF signal contributes only to the AC (random noise) component of a (mean = 0) Gaussian distribution, the expectation value of the square of the signal (the second-order absolute moment) is equal to the variance of the distribution.
In the limit of a large number of photons, the Poisson statistics of the LO approximate Gaussian statistics, and the variance of a convolution of Gaussian signals is equal to the sum of the individual signals’ variances. Thus, the other contributions to the variance of the signal (e.g the other noise sources mentioned above) can be measured when the thermal source is excluded from the photodetector (a “dark” measurement) and subtracted from the variance measured with the thermal source included. In terms of photocurrent, each variance represents the (electrical) power of the noise of each signal, and so noise power is additive in this mean-squared sense.The antenna theorem [8] gives a fundamental limit on the size of an aperture available for heterodyne detection by a single receiver:
where Ω is the receiver’s acceptance solid angle and S is the area of its aperture. This relation was derived from the requirement of spatial coincidence between the (single) laser mode and a single mode of the source waves and would limit the amount of heterodyne IF signal from a single aperture receiving both the laser and the source radiation regardless of the actual area and acceptance angle of the aperture being used. For this experiment, however, the source radiation intensity is derived from the additional Gaussian noise imposed on the detected signal by the random phase and intensity fluctuations of the thermal radiation source. It follows that addition of another heterodyned receiver signal produced by similarly captured radiation (uncorrelated to that captured by the first receiver) would add additional noise power as measured by the increase in the variance of the convoluted signal. As pointed out in [9], “… noises are incoherent, and the total noise power is proportional to the sum of the squares of the currents”. As the heterodyne IF current in each detector is noise, the total power is proportional to the sum of the squares of co-added heterodyne IF currents which, as shown above, is the sum of the variances of those photocurrents. The sum of apertures of multiple heterodyned receivers is not restricted by the antenna theorem, and so multiple receivers can increase the detectability of a given source in analogy with optical telescope apertures.Here, we demonstrate a technique where two heterodyne IF signals from the same broad-band, spatially extended, incoherent source (the sun) were convolved using a difference amplifier and produced a signal whose variance was equal to the sum of the variances of the individual signals. In this experiment, the balancing detector for the LO signal was used as the second heterodyne receiver, and so the heterodyne IF signal was increased without any additional sources of noise compared to the original experiment. Although adding even more receivers potentially adds other sources of noise (particularly more LO shot noise), the averaged noise from those sources can be removed by subtracting the results of the same measurement performed while excluding the thermal source (a dark measurement).
2. The instrument
A block diagram of the experimental setup is presented in Fig. 1. The setup includes the laser as the local oscillator (LO), an optical isolator (OI), variable optical attenuators (VOA), single mode optical fiber of type SMF28e, and single-mode fused fiber couplers (FC). The solar-collecting fibers (SMF1 and SMF2) were coupled with fiber from the LO and connected to the photo-detection block consisting of a balanced pair of photodetectors (BD1, including an amplifier and an electronic filter) and a Lecroy HDO 12-bit digital oscilloscope. Data from a reference channel included combined transmission signals from a low-pressure cell with methane (RC) and a fiber Fabry-Perot (FFP) interferometer on each side of a second pair of balanced photodetectors (BD2). For some data acquisitions, the FFP detector was disconnected to enable the full reference cell waveform to be captured and compared to theory. The essential difference between this instrument and that described by Rodin [6] is the addition of the second solar-collecting fiber (SMF2) applied to the balancing arm of the BD1.
We used a tunable distributed feedback laser from Eblana Photonics operating at λ = 1.6537 μm as the LO for these experiments. Broadband radiation from the sun passed through the atmosphere and was captured into fibers SMF1 and SMF2 by aspherical lenses with ∅1 inch, f = 50 mm and f = 87 mm, respectively, on a single solar tracking platform. The solar radiation was limited by passband filters ~19 nm wide, centered at 1.65 μm. The solar tracking system was connected to the instrument housed in the laboratory by ~20-m lengths of single-mode SMF28 fiber. Angled physical connectors (APC) were used to avoid reflected signals.
The balanced detector BD1 and signal processing are shown in detail in Fig. 2. BD1 included a trans-impedance (TI) amplifier (gain G1 ≈47,000) followed by an electronic bandpass filter operating from ~0.5 to ~10 MHz. The output of the bandpass filter was further processed by the digital oscilloscope: it was AC coupled to the scope, low-pass filtered, squared, and then scan-averaged.
Fig. 2 Details of the signal processing. The bandpass filtered signal from BD1 was AC-coupled, squared, and scan-averaged over the integration period.
An alternative arrangement was also investigated wherein the output of BD1 was input to a second stage amplifier (Stanford Research Systems SR445A) before being processed by an amplitude detector (AD) as shown in Fig. 3. This signal-processing arrangement is comparable to that of Rodin, et al. [6], though that paper does not give a detailed description of the amplitude detector circuit. The second-stage amplifier provided a gain and offset (G2~28; O~100mV) to the signal, and the AD consisted of a rectifying Schottky diode followed by a low-pass RC filter. The 100-mV input offset presented to the Schottky diode placed the bulk of the AC signal from BD1 into the square-law region of the diode’s current-voltage characteristic curve. The value of R2 (270 Ω) and C (10 nF) provided an integration time constant of τ = 2πR2C ~17 µs. The output of the AD was DC-coupled to the digital oscilloscope where it was digitized and scan-averaged. The advantage of this technique over the circuit in Fig. 2 is the reduction in high-frequency noise and lower-speed data acquisition requirements afforded by the low-pass filter as would be useful in a compact, limited memory, portable spectrometer.
Fig. 3 An alternative arrangement for the signal processing incorporated an “Amplitude Detector” (AD) whose output was sampled, averaged, and recorded directly by the digital oscilloscope.
The wavelength of the LO was controlled by a Thorlabs CLD 1015 laser diode controller with an input saw tooth voltage waveform to modulate the diode laser current. The sawtooth was repeated at a frequency between 200 Hz and 2 kHz with 500 Hz being typical. The controller also held the LO at a constant temperature, typically ~22.5 C.
Each input channel to the digital oscilloscope was bandwidth-limited to 20 MHz and was digitized at 25 mega-samples per second (MS/s). This sample rate represented large oversampling of the data, especially in the case of the AD output signal, so that smoothing (windowed averaging) was used to eliminate high-frequency electronic noise during post processing. Although the signal repetition rate was between 200 Hz and 2 kHz as stated above, the digital oscilloscope averaged scans at a much slower rate because of signal processing overhead. Therefore, for example, averaging 4000 scans at 500 Hz (i.e. eight seconds of signal integration time) took nearly two minutes.
Wavelength calibration was achieved by using a methane absorption feature from a 10-torr CH4, fiber-coupled reference cell (Wavelength References, Inc.) as an absolute wavelength reference, and interference fringes generated by the fiber Fabry-Perot in the reference channel to determine the change in wavelength as the diode current was scanned. The Fabry-Perot fringes were calibrated to wavenumber by modeling the methane absorption peak width using the GENSPECT line-by-line radiative transfer code [10] with the HITRAN2008 spectral database [11] and the reference cell fill pressure, temperature and pathlength.
3. Data Processing
In order to elucidate the results, the following will demonstrate the method used to derive the source radiation power spectrum from the signals generated at the receivers of the balanced detectors
Figure 4 shows the amplified output of the BD circuit at trans-impedance output TP1 as laser current is increased as a function of time. The two signals could not be exactly balanced at all LO currents possibly due to variations in the transmittance of the fiber and fiber couplings and in the detectors’ responsivity as a function of LO wavelength. Therefore, a low-order polynomial fit to the TP1 signal (shown in red) was used to eliminate the low-frequency components of the signal. The difference between the real-time signal and the polynomial fit represents the higher-frequency AC components of the signal and is also shown. In normal use, however, the AC components are measured directly after the frequency bandpass filter at the output of BD1. The square of this AC signal is proportional to its (electrical) power within the frequency bandwidth of the detector/amplifier and is shown in Fig. 5.
Fig. 4 The output of the BD1 amplifier at TP1 during a single scan of the laser (blue) and a polynomial fit to it (red). The difference between the two represents the AC components of the signal within the frequency pass band of the electronics.
Fig. 5 The AC signal of Fig. 4 is squared (black) and smoothed with a 100-data point moving average. The smoothed data demonstrates an increasing noise magnitude primarily due to the increase in the LO laser power which increases the laser shot noise.
At a given laser current (wavelength), the magnitude of the AC components will fluctuate from scan to scan. Averaging many scans can produce a spectrum of the average squared noise amplitude (Eq. (3)) which increases with LO laser power as laser shot noise dominated the noise signal. Alternatively, averaging over small sub-intervals of time (i.e. data-point smoothing via a moving average, also shown in Fig. 5) can also produce an averaged squared noise amplitude, albeit over the wavelength interval corresponding to the number of data points used in the smoothing. As described previously, combinations of both averaging techniques were used, i.e. scan averaging (time integration) and data smoothing (spectral averaging). Residual noise is then proportional to both the square root of the integration time and the square root of the number of data points in the smoothing window. It is this residual noise that is used to calculate the final signal-to-noise ratio (SNR).
4. Observations
Here, we demonstrate the advantage of the multiple receivers by comparing solar radiation spectra taken with each of two single receivers to that taken with both receivers simultaneously. Though atmospheric absorption spectra are derived from the measurements, there is no attempt in this paper to analyze the spectra or explain their features.
Figure 6 shows the averaged, squared BD1 output signal (Fig. 2) under four conditions: The “Dark” condition was obtained when both of the solar radiation-receiving fibers were disconnected. “F1 (+)” and “F2 (−)” denote the conditions where only one solar receiver was connected, that being either SMF1 or SMF2, respectively connected to the ( + ) or (−) photodetectors of the BD1. “Both” refers to the condition where both receivers were connected as shown in Fig. 1. The data were taken in the order shown in the legend. The difference in the signal levels from the two fibers is primarily due to differences in detector response. (That is to say, swapping the fibers leads to similar signal levels from each of the two detectors.)
Fig. 6 The signals under four conditions using an 8-second integration time after a 40-point, post-processing smoothing procedure. “Dark” refers to the condition of no solar input while F1 and F2 denote single-fiber solar inputs. “Both” refers to the condition of simultaneous solar input from both fibers.
The two ‘Dark’ data sets were then averaged, and the result was fitted to a fourth-order polynomial to yield a baseline signal that was subtracted from all the signals. The results shown in Fig. 7 clearly indicate that the sum of the two single-fiber signals approximates the single two-fiber signal well within the noise. Figure 7 also shows that while the signal increases, the noise stays relatively constant, and thus there is a net gain in the signal-to-noise ratio (SNR). Note that adding the squared individual signals’ variances arithmetically does not afford the same benefit in SNR as measuring the variance of the electronically mixed signals. In the former case, one expects the SNR to increase as the square root of the sum of the squares, i.e. (1.72 + 2.62)1/2 ≈3.1 as shown. The SNR of the mixed signals case is much closer to the sum of the SNR of the individual receivers: 1.7 + 2.6 = 4.3 compared to the measured value of 3.8.
Fig. 7 The signals after 8 seconds of integration, a 40-point smoothing, and subtraction of the mean dark signal. The red curve represents the arithmetic sum of the two single-fiber signals (light and dark blue) and is closely overlaid by the signal with both fibers connected (dark green). The black traces are the fiber signals after further smoothing (150-pt window, applied 50 times). Noise as represented by the 3-σ deviation of the signals from the black traces was used in determining the SNR ratios shown in the figure which pertain to the 40-pt-smoothed curves.
The results demonstrate that the noise generated by the heterodyne IF component of the photodetector current when squared is additive as variance would be additive for Gaussian noise from multiple receivers, thus increasing the minimum detectable solar radiation. This can enable decreased integration time and/or an improved signal-to-noise in the mean-squared amplitude for a given dark-corrected measurement.
The same increase in signal was also demonstrated using the amplitude detector circuit (Fig. 3) similar to that of the original work [6]. When this configuration was used, however, overall signal drift was noticeable on timescales needed for signal integration. This drift was limited by performing measurements rapidly between the single-fiber and the dual-fiber configurations. Such a result is shown in Fig. 8. Here again, the dark signals were averaged and fitted to a low-order polynomial which was subtracted from the data. The drift in the dark signals is evident, but the summation of the single-fiber signals reproduces the dual-fiber signal well within the drift error.
Fig. 8 Successive measurements taken with the AD circuit shown in Fig. 3. Measurements (including the “Dark”) are shown after 40-pt smoothing and subtraction of the mean dark signal.
5. Atmospheric transmittance spectrum
An atmospheric transmittance spectrum can be estimated from the solar radiation spectrum by assuming the solar continuum is a linear fit to the data when obvious absorption features are ignored. Such a spectrum in shown in Fig. 9 which generated using data averaged over 85,000 scans at 500 Hz (approximately 170 seconds of integration time, over about 15 minutes of real time). This achieved a 3-σ detection level of approximately 4% absorption.
Fig. 9 An atmospheric transmittance spectrum obtained by fitting a continuum spectrum to a dark-subtracted spectrum as shown in the inset graph. The spectrum was obtained on the same day as those shown in Fig. 5. As indicated, high clouds were present in the atmosphere when the data was taken.
The spectrum is compared to a model spectrum based on the HITRAN database simulating only methane, carbon dioxide and water molecules. The assumed profile is based on column concentration profiles from mid-latitude measurements [12] and pressure-temperature conditions of a standard atmosphere.
Though a detailed spectral analysis will not be given here, of note in this spectrum are features that are not found in the HITRAN 2008 database and have not yet been identified. There is a very consistent feature at ~6047.2 cm−1 next to the methane absorption at 6046.96 cm−1. Underlying both absorption features seems to be a broad feature extending from 6046.2 to 6047.4 cm−1 (see Fig. 9). The combined H2O/CO2 absorption at ~6047.8 cm−1 has possibly another broad absorption at its low-frequency shoulder. These broad features could either be experimental artifacts due to instability in the electronics, solar features, or atmospheric absorptions. Of note is that subtraction of the absorption underlying the methane absorption would increase the minimum transmittance of the methane line to about 40% where the theoretical calculation predicts 20%.
6. Conclusion
The feasibility of improving the signal-to-noise ratio of data taken using the high-resolution NIR heterodyne detection system previously described by Rodin, et. al [6], has been demonstrated using a slight modification of the original experimental apparatus. In the original experiment, two detector signals were differenced to balance the common mode noise of the laser intensity while the heterodyne IF signals were created by mixing sunlight with the laser on only one of these detectors. By mixing sunlight with laser light on each detector individually, and then on both detectors simultaneously, this work has shown that the thermal source power in the balanced signal could potentially be doubled. That is to say, the sum of the squares of the two single-detector heterodyne IF signals was shown to be equal to the overall heterodyne IF signal squared produced when both detectors were used for photo-mixing (after subtraction of the laser-only measurement). Since the spectral power density of the thermal source was proportional to the square of the heterodyne IF signal, the measured SNR of the source spectrum was increased by nearly a factor of two compared to the single-detector photo-mixing results. Thus, photo-mixing on both sides of balanced heterodyne detectors maximizes the amplitude of the heterodyne IF signal components without increasing any other noise components.
Further to that, the results demonstrate that the ability to add uncorrelated noise signals as variances provides the means by which multiple apertures can be used to increase total heterodyne IF signal throughput beyond the limit imposed by the antenna theorem on a single aperture of equivalent étendue in the case of a spatially-extended thermal source.
Acknowledgements
This work was funded by UNSW-Canberra Space under Professor Russell Boyce. Thanks also to Assoc. Prof. Andrew Lambert for helpful discussions.
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