1. Introduction

The advantages of detecting discrete transitions through frequency-modulation spectroscopy (FMS) are well known and include good rejection of spectrally-broad background absorption and reduced noise brought by phase-sensitive detection of the FMS signal at often high frequencies of roughly the transition linewidth [1,2]. As we will show, the technique of code division multiple access (CDMA) partners naturally with FMS to add the capacity of multiplexing, yet the two had not been combined previously, to our knowledge. While FMS techniques have flourished in applications of diode-laser spectroscopy, the principles of CDMA multiplexing apply quite generally to FMS over its broad spectral practice, ranging from the microwave [3,4], the mid- and near-infrared [5–8], to visible transitions [9–11], and the ultraviolet [12–16]. Under CDMA, transitions, whether from different species, isotopes, or the same absorber, are assigned unique codes, and codes may arrive at the receiver at various lag times. Incident on the single receiver, then, is a multiplexed signal carrying detection composition from all encoding delays.

The field of digital communications gives our approach context and presents a far richer source for ideas and variations beyond the basic workings we outline and demonstrate here. Our approach to multiplexing falls, as stated above, within the general category of code division multiple access, a spread-spectrum technique long used in radar systems and military communications [17] before becoming commonplace in digital wireless communications. In a basic implementation, CDMA transmits all channels at the same time, assigning to each a pseudo-noise (PN) code that is uncorrelated with any other. At the receiver, the total signal is cross-correlated with those individualized codes [18] to reveal, from what appears to be indecipherable noise, the contents and lags of the associated channels. Originally developed to withstand jamming, CDMA and the process of cross correlation is very effective at suppressing narrowband noise and interferences [17,19,20], as might be present on the signal or introduced by the receiver. Further, the signal-to-noise ratio of a CDMA code, which will come to represent our measurement packet, increases in proportion to the code length irrespective of how noise (of finite power) is distributed over the bandwidth carrying the code [20].

In the case of light detection and ranging (LIDAR), the use of a continuously-repeated PN sequence for ranging seemed to appear first as a form of pulse-code modulation (PCM) [21], for example when Takeuchi et al. in 1983 [22] directed the pulse-modulated output of a 1-W Ar⁺ laser vertically to measure the nighttime aerosol-layer distribution above Tokyo. The single PN code they transmitted brought a multiplexed response from the distributed target, consisting of a superposition of the same code at various delays and amplitudes. Decoding converted the raw multiplexed return into the signal by range, at the limiting resolution set by the rudimentary bit or “chip” interval in the PN code [17,22]. The extension of PCM [22] to two-wavelength differential absorption LIDAR (DIAL [23,24]) for purposes of species detection eventually followed [25]. In 2004, Bashkansky et al. [26] adapted LIDAR to take advantage of phase-modulation techniques, by imprinting an RF carrier with a pseudo-random binary phase-shift key (BPSK) onto the intensity modulation of continuous-wave beam, which they used to range hard targets. Here, we similarly combine phase modulation and BPSK’s but place them within FMS, extending a CDMA capacity to that spectroscopic technique that can be put toward ranging or multi-channel (e.g. multi-species) monitoring, or both. Readers will find ample literature on the signal analysis and applications of FMS since its initial description by Bjorklund, Levenson, and coworkers in 1983 [2].

FMS is an optical heterodyne technique, where phase modulation at angular frequency ω_m is impressed on the carrier of ω_c >> ω_m of a single-frequency laser beam, creating a set of sideband frequencies in the electric field that can produce a beat note at ω_m at a square-law detector after the beam passes through a dispersive medium. Homodyne, phase-sensitive detection of the beat-note signal produces in-phase (X) and quadrature (Y) components of

In Eqs. (1) and (2), a₁ and b₁ are taken from our recent modeling [27] of FMS, defined there as, respectively, the cosine and sine coefficients of the signal’s component oscillating at ω_m. Angle ϕ is the phase angle imposed during a bit interval of a phase-shift key. Figure 1 illustrates X and Y as projections of R onto axes x and y in response to ϕ that rotates the subsidiary axes (of x′ and y′) defining a₁ and b₁. For sake of generality, we retain in Fig. 1 the arbitrary phase angle of phase-sensitive detection θ, per Ref. [27], as distinct from the intentional application of ϕ. In a BPSK sequence, bit intervals are assigned ϕ = 0 or π, constraining X and Y of Eqs. (1) and (2) to

 

Fig. 1. Diagram showing the separately-applied phase angles that set the projection of signal-magnitude R onto axes x and y, producing the in-phase (X) and quadrature (Y) components. In the frame (x′,y′), angle θ tunes the ratio of the cosine (a₁) and sine coefficients (b₁) composing R. Angle ϕ then rotates frame (x′,y′), yielding X and Y by Eqs. (1) and (2) in general and by Eq. (4) for the 0–π phase alternation of a BPSK.

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While BPSK waveforms are typically monitored by the in-phase component and with θ set for |X| = R and thus Y = 0 [Eq. (3)], there is no guarantee in this application that a phase angle that nulls Y in one channel will similarly null Y in another, with simple examples drawn from assigning channels to spectral lines with each a different ratio of a₁/b₁. In our experiments we measured both X and Y set roughly equal through minor adjustments of θ, giving our bipolar signal of demodulation, S, as R signed by either X or Y, or

2. Approach

Figure 2 shows the approach. The output from a continuous-wave, single-frequency laser operating at ω_c is conditioned by a phase modulator (the electro-optic modulator, or EOM) driven at RF frequency ω_m << ω_c (provided by RF osc.) at a power corresponding to modulation index δ (not indicated in the figure), producing a spectrum in the outgoing light consisting of ω_c flanked by sidebands at intervals of ω_m as depicted generically by the inset graph. The process of electro-optic phase modulation and the resulting optical spectrum are well known [28]. Formulas and graphs for setting ω_m and δ to yield strong FMS signals from common absorption line shapes are also available [27,29]. In a typical FMS experiment, the beam transits an absorption feature of width ∼ω_m near ω_c and terminates on a detector to provide homodyne signals X, Y, and/or R as described in Eqs. (1)–(3). Here, however, we imprint a BPSK on the outgoing laser spectrum. Before being fed to the EOM, the driving RF passes through a bi-phase modulator (see Fig. 2) that converts a TTL bit stream of the unipolar {0, 1} key at its control input to the corresponding BPSK sequence ϕ(t) = {0, π} of phase additions to the RF. The BPSK is mapped from the bipolar {-1, 1} key M(t) = cos[ϕ(t)], which we take to be a maximum length binary sequence (MLS) [30] of period t_p = Nt_b consisting of N bit intervals of duration t_b >> 2π/ω_m. From the EOM, in response to ϕ(t), the laser spectrum switches to and from having its odd-order sidebands rotated by a net phase of π radians [27], an alternation that becomes the source of sign changes to X and Y of Eq. (4). At the detector and after phase-sensitive detection we then find S(t) [Eq. (5)] to be, theoretically, a scaled copy of M(t) delayed by a transit time T. Both M(t) and S(t) are sampled at uniform intervals t_k = t_b/k, where k is a positive integer [31]. The normalized cyclic cross-correlation $r(\tau)=M \otimes S /(k N)$ then yields, in the ideal as illustrated in Fig. 3, a triangular peak at lag τ_pk = t_k×nint(T/t_k) [32] of height |S| and base 2t_b from a floor of −1/(|S|×N) [33–35]. In practice, the signal is conditioned by a low-pass filter of bandwidth B set to encompass most of the spectrum of M(t) described itself by a sinc²{(t_b [s])×(f [Hz])} envelope, leading to B lying usually within 0.5 $\mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $ (t_b [s])×(B [Hz]) $\mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $ 1 [35–37] and a sampling rate 1/t_k = k/t_b set commensurately from one to several times the Nyquist rate of 2B. The finite bandwidth has the net effect of limiting the cross-correlation signal to a resolution of 2 $ \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $ τ/t_b $ \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $ 1.2 for distinguishing peaks of equal height [38], with the finer resolution approaching the full-width-at-half-maximum (FWHM) of τ/t_b = 1 of the ideal peak. In the case of free-space LIDAR, where the transmitter and receiver roughly coincide, the collection of returns from discrete or continuous targets would thus have a spatial resolution Δ_R, cast by Δ_R/t_b, of 300 to 180 m/µs, this again in consideration of a finite B.

 

Fig. 2. Operation concept of multiplexed FMS in the context of LIDAR, where a pseudo-random phase-shift key (the alternating sequence in black) imprinted on the laser transmitter is recovered as a delayed copy (blue sequence) of alternating signal sign at the receiver following light detection by a square-law detector (Det.) and demodulation. See text for details.

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Fig. 3. Idealized cyclic cross-correlation, r(τ), between N-bit key M(t) and signal S(t), having been sampled in this illustration at intervals t_k of half the bit duration t_b. Primes for temporal symbols indicate normalization by t_b.

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Additional beams tuned to detect different transitions may be added to the transmitted beam, with each being assigned a unique BPSK M_i(t) clocked at bit rate 1/t_b in the manner shown in Fig. 2 by a bi-phase modulator and an EOM of adjustable modulation depth driven at ω_m. The detector and demodulator require no change. Cross correlation of total signal S(t) by each M_i(t) reveals the lag spectra of the contributing S_i(t). Different pseudo-noise keys, however, in their cross correlation do not produce the flat autocorrelation baseline shown in Fig. 3 and so require care in choosing to keep their cross-correlation spectra low and the distinguishability of their associated signals high. Fortunately, there are a number of PN and PN-related sequences to choose from [18,30], including families of sequences specialized for low cross correlation [18]. Our numeric tests showed that good PN pairs are easy to find and need not have equal periods. Superposed, distinct, and equal-amplitude MLS pairs, for example, appeared each per Fig. 3 as sharp peaks in decoded lag spectra and above a cross-correlated noise baseline of σ ≤ 1% for a least N of ≥8191. The technique overall needs only two, independently operable clocks. Frequency modulation and demodulation operate by a single oscillator frequency (ω_m), and the bit rate 1/t_b of BPSK encoding and decoding is set by a separate clock. Tying the modulation/demodulation and encoding/decoding processes to their single clocks affords to each, and to the final decoded signal, a good immunity to noise from drifts in ω_m and 1/t_b. At the receiver stage, the de-multiplexing process acts on the direct, undivided receiver signal, needs only a single demodulator, and applies to modulation from low frequencies through the typically ω_m/2π ∼ GHz range of FMS. This approach offers the signal-path and signal-analysis simplicity of the frequency-encoding/digital-decoding schemes favored for the comparatively low-frequency operations of wavelength modulation spectroscopy [6,7] but at the higher modulation frequencies more characteristic of FMS.

3. Experiments

Our lasers and optical components targeted the infrared transitions of CO₂ in the 30012←00001 band centered near 1.5753 µm [39], which is among those used for remote-sensing measurements of atmospheric carbon dioxide [40]. Laboratory validations preceded our outdoor LIDAR trials, beginning with a straightforward comparison between measured and modeled FMS spectra followed by an experiment using an optical-fiber delay line to simulate ranging in a controlled environment.

Figure 4 shows an abridged diagram of the laboratory setup. A grating-tuned diode-laser system from New Focus served as the laser, and the EOM was a fiber-optic modulator from JDSU (Model # 21010696). With reference to Fig. 2, our RF oscillator (Exodus Dynamics EDRO-1000-01.50) was fixed at ω_m/(2π) = f_m = 1.5 GHz. The bi-phase modulator (M-1250-BPSK) placed in the EOM drive was from ET Industries, and the I/Q demodulator (QDO-622B) was from Polyphase. Light propagated within polarization-maintaining fibers from the output port of the laser to the detector (Det) except for free-space transmission through the 85-cm-long absorption cell filled with industrial-grade (99.5% purity) CO₂ to typically 69 kPa (520 Torr). A 2-km coil of fiber placed after the EOM received roughly 5% of the light in the main fiber, providing a delayed return of distinctly different amplitude from the direct path to the detector. The recombined beams also had orthogonal linear polarizations, which minimized their optical interference at the detector. Elements of the setup not shown include diagnostic tools of fiber and free-space Fabry-Perot interferometers, which we used to gauge the frequency-tuning rate of the laser during spectroscopy experiments and to measure the modulation index delivered by the EOM.

 

Fig. 4. Simplified diagram of the laboratory setup, which allowed for coupling ∼5% of the light into a delay line made from a 2-km coil of fiber, simulating a LIDAR return. The room-temperature cell had an 85-cm absorption path and was filled with CO₂ at typically 69 kPa (520 Torr).

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To check our FMS model [27], we measured the FMS profile of P(14) of the 30012←00001 band of ¹²CO₂ [39] with the delay line removed and for various modulation depths. This is a line among those accessible by our laser with good strength at moderate temperatures and minimal interference from water-vapor spectra over the longer paths planned for outdoor experiments [41]. Figure 5 shows two example FMS profiles, one for a measured modulation index of 0.58 and the other for 3.5 (∼15 Vp-p to the EOM). For both cases, the cell was at room temperature (296 K) and filled with CO₂ to 523 Torr. Solid lines are the modeled profiles of magnitude R [Eq. (3)] versus laser frequency, which in addition to δ and f_m incorporated the P(14) line strength and self-broadening parameter from Boudjaadar et al. [39] and Toth et al. [42], respectively. The two measured profiles (open circles) were shifted horizontally and then scaled vertically by the same factor for best match with the model. As seen, the model captures the measured profiles quite well, including the 2.2× increase in peak magnitude on raising δ from 0.58 to 3.5. We proceeded with δ set to 3.5 for our laboratory and outdoor ranging demonstrations, as our models further showed this value sufficed to provide peak magnitudes within 0.5% of the maximum possible (for our fixed f_m) under those conditions.

With the delay line inserted, and the laser frequency coincident with the peak FMS magnitude, we produced cross-correlation spectra as outlined in the previous sections. The cross-correlation result shown in Fig. 6 used a 1023-bit MLS key [30] delivered at one bit per microsecond (t_b = 1 µs), corresponding to bit intervals of l_b = 210 m within the fiber [43]. The graph shows the decoded signal to have the expected isolated peaks at ranges of zero (for the direct path) and 2 km. The 23:1 ratio between peak heights is in accord with that expected by the ∼5% split of light power to the fiber loop, demonstrating that the decoding process recovers the strength as well as the range of the superposed encoded returns. Each peak appears triangular in shape with a base of 2l_b, as expected. The yet smaller replica of the cross-correlation peak discernible at twice the fiber-coil delay of 4 km is consistent with a second trip through the coil brought by weak reflections from the fiber splitters used to splice the delay line into the circuit.

 

Fig. 5. Measured and modeled FM spectra for large modulation depths, of the CO₂ P(14) 30012←00001 transition recorded across the cell of Fig. 3 filled with CO₂ to 69.7 kPa (523 Torr).

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Fig. 6. Measured cross-correlation signal for the laboratory setup of Fig. 4, where the time scale has been converted to propagation distance through optical fiber. The fiber-delayed peak occurs at the expected strength (5%) and distance (2 km) relative to the peak produced by the un-diverted light, centered about zero. The smaller peak discernible at about 4 km is consistent with a second pass through the delay line brought by weak reflections from its associated fiber couplers.

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The outdoor experiment used a horizontally-directed beam and followed the diagram of Fig. 2 with the addition of a single-mode fiber amplifier (IPG Photonics) placed after the EOM and operated at 2.5 W. Conditioning optics collimated the beam to a diameter of roughly 30 cm at a distance of 150 m. A 10-inch-diameter (25 cm) hobbyist Cassegrain telescope collected the backscatter and focused it onto a high-speed InGaAs avalanche photodiode that provided a high sensitivity to the expected weak levels of collected light. As diagrammed in Fig. 7, the transmitter passed through two “gas plumes” represented by 73-cm-long by 35-cm-diameter acrylic cylinders filled with neat CO₂ to about 79 kPa (or 590 Torr, the average atmospheric pressure at Los Alamos). Directly behind each cell we placed scattering targets of aged, oxidized aluminum sheets. At the first cell, the target’s occlusion of the beam was partial and adjusted to obtain roughly equal levels of received light power from both targets when the cells contained only ambient air. The two backscattered light signals were found to be additive, signifying that our optical setup was far from the ideal needed to produce a net interference between the returns at the detector. Our field-tape measurements to the cells are indicated in Fig. 7 and showed the scattering planes of the near and far cells to be 65 m and 173 m, respectively, from the convenient reference position of the receiver. Because the area was frequented by other workers during the day, safety considerations had us conduct our experiments at night when daylight interference was absent as well, obviating any need for optical filtering. Column densities of atmospheric CO₂ along our paths were, for our purposes, a negligible ∼5% of that presented by the cells.

 

Fig. 7. Diagram of outdoor experiment. The laser and BPSK components diagrammed in Fig. 2 are augmented here with a single-mode fiber amplifier placed after the EOM to form the laser transmitter, which delivered a 2.5-W collimated beam ∼30 cm in diameter. Two sealed, gas-fillable acrylic cylinders, each 73-cm long and 35 cm in diameter, were placed in the beam path to represent gas plumes and stimulate FMS signals when filled with CO₂. Oxidized aluminum plates behind each cylinder served as scattering targets. As drawn, the targets were spaced 108 m apart, with the near target set 65 m from the receiver and occluding only part of the beam. The labeled 10” Receiver consisted of a hobbyist telescope with a high-speed avalanche photodiode at the focal plane. LO is the local oscillator, equivalently the RF oscillator of Fig. 2.

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Figure 8 shows our ranging measurement obtained from a 1000-period average (covering <5 s) of a 46691-bit QRB key [44] delivered at t_b = 100 ns and filtered by B = (t_b)⁻¹ = 10 MHz at the receiver. To account for electronics delays, we translated the cross correlation to set the first peak at the reference distance of 65 m, from which the second peak is seen centered 108 m away at 173 m as expected. The FWHM of both peaks of 16 m is also as expected for our 10-MHz bandwidth (which, by the relationships given under Approach, would produce a spatial resolution Δ_R = 18 m). In this controlled exercise, the peak heights can be used to estimate the relative light powers in the two returns in the absence of the absorber. Our measured peak ratio of far/near of 1.53 was shy of the 1.85 we calculated for perfectly balanced returns, thus indicating we started with ∼20% more light along the roundtrip to and from the near cell. We attribute the baseline noise to various non-idealities in our components that, while adequate for our demonstrations, could be upgraded.

 

Fig. 8. Cross-correlated LIDAR measurement for the setup of Fig. 7 with the transmitter delivering a 46691-bit key at 100 ns per bit. The data, shown in circles, are from a 1000-period (4.7 s) average of the return signal; the line is a splined fit through the points added only as a visual complement. The vertical scale is normalized to the height of the expectedly-stronger far peak, and a circular translation of the cross correlation set the near peak at the known range of 65 m to account for unknown electronic delays. In accord with Fig. 7, the two peaks are separated by 108 m. The 10-MHz filtering of the receiver signal rounds and broadens each peak slightly from the triangular ideal of 15 m FWHM to one with 16 m, in good agreement with expectations.

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4. Conclusions

We have shown how to augment FMS with well-known CDMA techniques, through the application of binary phase-shift keys, to produce a multiplexed capacity. Laboratory experiments validated the experimental transcription of CDMA to FMS, where the amplitudes and phases (arrival times) of overlaid codes were correctly extracted. We then demonstrated the multiplexed aspect in a prototypical outdoor LIDAR set-up involving simultaneous, continuous returns from two distant scattering surfaces each preceded by a short-length cell containing the target gas, which showed that a multiplexed ensemble decodes properly into ranges and their FMS-signal amplitudes. Our multiplexed approach can be applied wherever FMS is suitable. Given practical limits on the available levels of power from continuous-wave lasers, the application of this approach to continuous free-space ranging measurements is likely realistic only against strongly scattering media and hard targets, including, as an extension of Figs. 7 and 8, a series of small reflectors defining spatial intervals of monitoring. Aerosol and droplet clouds are suitable backscattering targets for FMS [45] and also, as introduced earlier, for density-profile mapping by a pseudo-noise PCM sequence [22]. Under a CDMA approach, pseudo-noise FMS codes and a PCM code (set to a laser tuned off-resonance) could together reveal species profiles corrected for Mie-scattering attenuation. The PCM sequence could alternate with the FMS codes, or all could be transmitted simultaneously, the latter instance capitalizing on the ability of PN codes to reject uncorrelated noise. The extension of FMS to pulsed-laser sources [12,13] gives rise to extra prospects for this multiplexed approach, where the bit interval and frequency would be set by the pulse duration and repetition rate, respectively. In the most general instance, keys are associated with transitions, and while we used in these initial experiments a single key, multiple pseudo-random sequences can be overlaid with each exhibiting good distinguishability in cross correlation [17,30], leading to a rich multiplexed construct where multiple transitions and ranges are monitored simultaneously.