Differential absorption lidar system sensitivity with heterodyne reception B. J. Rye

University of Hull, Department of Applied Physics, Hull, HU6 7RX, United Kingdom. Received 6 June 1978. 0003-6935/78/1215-3862$0.50/0. © 1978 Optical Society of America.

The use of wavelengths other than the uv or visible wave­lengths currently used for incoherent backscatter differential absorption lidar (DIAL) systems has the potential advantage of greatly increasing the number of chemical species that can be detected.1 In the ir, heterodyne reception has the further advantage of immunity against background so that the min­imum detectable signal is less than with direct detection, and a better range capability might be expected for a given laser energy.2 While the practical difficulties of using the hetero­dyne technique at long range, such as the need for accurate and stable alignment of the transmitted and local oscillator beams, are probably generally recognized, the signal pro­cessing requirements, which are different from those of a di­rect detection system, do not appear to have been discussed adequately in the literature in this context.

The principles1 of the DIAL technique considered here are that the attenuation of a transmitted laser pulse over a range r can be inferred from measurement of the return photon flux +ΦR (r), provided by incoherent backscatter from molecules and distributed aerosols and dust particles in the atmosphere; if pulses are transmitted at two frequencies 1 and 2, respec­tively, on and off the absorption resonance of a particular molecular component of the atmosphere, the fractional con­centration (e.g., in ppb) of the species over the range resolution Δr can be inferred by comparison of the return fluxes using the equation1

when nn is the over-all number density of molecules in the atmosphere. The sensitivity of the measurement (i.e., the lowest variation in γp that can be resolved) is given by dif­ferentiation of this equation as

where (S/N) represents the over-all signal-to-noise ratio in the measurement of the term within the logarithm in Eq. (1). This Letter is concerned with calculation of (S/N) for a system employing heterodyne reception and operating under ideal conditions, so, for example, atmospheric scintillation and excess detector noise are neglected.

An expression for the observed signal power for this case is obtained in the reference quoted in Ref. 3. Incoherent backscatter gives rise at the receiver to a partially coherent field or speckle pattern; the return can be characterized sta­tistically by +ΦR, the mean number n1 of spatial modes in­cident on the receiving antenna aperture and the field corre­lation function gs() of the speckle pattern fluctuations in time. A heterodyne receiver is further characterized by the heterodyne efficiency ηh and the local oscillator photon flux +ΦL,incident on the photodetector, the quantum efficiency of which is taken to be η. For our present purpose it is con­venient to suppose that the ac coupled output from the pho­todetector is integrated over a sampling time d and to cal­culate the moments of the resulting sample charge q; the in­tegration corresponds in practice to what is done at the front

3862 APPLIED OPTICS / Vol. 17, No. 2 4 / 1 5 December 1978

of a digital processing system and also simplifies the equations. The second moment is given by3

where the first term is the (generation) shot noise of the local oscillator photocharge, and the second is the mixing term. The integral can be simplified by the further assumption that the difference between the local oscillator and scattered light frequencies ω = ωL – ωS = 0 and by introduction of a zonal gating function hd(t) to describe the sampling interval. Putting

the integral becomes

where gd() is the self-convolution of hd(t) normalized so that gd (0) = 1. We can then define the number of temporal modes observed in the integration time d (by analogy with the definition in Ref. 3 of n1) as

and rewrite Eq. (3) as

Here +NL, = +$ L , d , and

is the received photon count +NR, = +ΦR, d in the limit m1n1 = 1, while in the limit m1 » 1, n1 » 1, δh is a degeneracy parameter of the return.

It is now necessary to consider the method by which the +ΦR, terms contained within the logarithm in Eq. (1) can be determined so that an appropriate value of S/N can be derived for use in Eq. (2). In this discussion the variation of +ΦR(r), with range will be neglected as it is assumed that Δr « r.

In principle each of the +ΦR, are inferred by squaring and averaging appropriate sample values of q and making use of Eqs. (3) or (5). It is possible to show as follows that any sample values of q appropriate to different terms +ΦR, in Eq. (1) are necessarily uncorrelated:

(1) Consider, first, any two samples of q measured using the returns from ranges separated by Δr obtained with a single transmitted laser pulse; such samples contribute to the terms +Φ(r)

R(r), and +Φ(1)R(r + Δr) [or +Φ(2)

R(r) and +Φ(2)R(r + Δr),].

Now the fluctuations of the speckle pattern have a correlation time of approximately π/Δω, where Δω is the bandwidth of the pulse; the latter is related to the depth ρS of the scattering volume contributing to the speckle pattern at a particular instant of time by ρS > π(c/Δω), where the equality sign applies approximately if the transmitted pulse is transform-limited. Since the range resolution Δr > ρs, it follows that the delay between the two returns considered is 2Δr/c > 2Π/Δω, so that the speckle pattern varies within this delay

time. Fluctuations in the speckle pattern integrated over the sampling time d produce corresponding fluctuations in the sampled charge q. Since d 2Δr/c (so that returns from ranges separated by Δr are resolved) the required result fol­lows that sample values of q obtained from these returns are uncorrelated.

(2) Consider, second, samples of q measured using returns from a single range but obtained with transmitted pulses at two frequencies ω1 and ω2. It might be supposed that these values would be related if the two pulses were transmitted simultaneously so that the geometrical configuration of the scattering centers was the same for both. Here it is necessary to note that the field amplitude in the speckle pattern can be regarded as a random walk summation of the field produced by each randomly situated source within ρS; it is therefore only independent of frequency as long as the phase variation of the component fields is much less than π/2, i.e. (since the path differences from the scattering source to the receiver vary by as much as ps) over a spread of wavenumber Δk, where Δkρs « π/2. For our problem we have Δk = (ω1 – ω2)/c, where to avoid frequency overlap between the pulses ω1 – ω2 > Δω. Use of Δω > 2Ωc/ΡS as above then yields Δkρs > π so that sample values of q obtained from these returns are also uncorrelated; such samples contribute to the terms +ΦR(r), and +ΦR

(2)(r), or +ΦR(1)(r + Δr), and +ΦR

(2)(r + Δr), in Eq. (1).

The statistical independence of all relevant samples of q implies that each of the mean flux terms +ΦR, in Eq. (1) has to be determined separately from the related sample values of q. If the signal to noise ratio appropriate to measurement of each of the +ΦR, is denoted (S/N) i, we can therefore sub­stitute in Eq. (2)

In papers on incoherent backscatter lidar2,4 it has been customary to use the ratio of the two terms in Eq. (5) as the signal to noise ratio of the receiver, whereas it is in fact the ratio of the mean signal power to the local oscillator noise power in the photodetector output. In a system designed to measure signal power (or photon flux) this output is detected, and as is well known the signal to noise ratio can be signifi­cantly impaired on transmission through a nonlinear device.5,6

In the present case the final signal to noise ratio of the flux measurement is rather easily determined by using the result that the distribution of the sample values q is approximately a Gaussian of zero mean3 so that the variance of +q2, is

Noting from Eq. (5) that in practice a determination of δh can only be made by taking the difference between observations of +q2, and observations of +q2,L

= e2Η+NL, (with δh = 0), and adding the variances of these two measurements, we ob­tain for the final signal to noise ratio of a single sample

which has the limiting values

Thus (S/N)1 is proportional to δh only when 2ηηhδh « 1, i-e., when the signal to local oscillator noise ratio at the photode­tector output is much less than unity; this corresponds to the

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small signal suppression condition in the classical treatments6

and to there being an average of less than one detected pho-tocarrier per observed mode per sample in the present context; using Eq. (5) it can be seen that (S/N)1 is only proportional to the return flux in the small signal suppression regime and if in addition m1n1 = 1. The saturation of (S/N)= at high re­turn flux (2ΗΗhΔh » 1) occurs because the predominant noise arises from fluctuations of the return speckle pattern. Similar phenomena have been described in the context of heterodyne receivers employing analog processing.7,8

The signal to noise ratio appropriate to a particular value of +ΦR(r), can be improved by taking more samples within the given range resolution element. In fact analog processing corresponds to taking such samples continuously, in which case the final results depend in detail on the autocorrelation function or the frequency spectrum of the return. This complication, which introduces to the results factors of order unity, can be avoided without significant loss of generality in the present context if it is supposed that independent samples are taken. We then have (S/N)i = (mnM)1/2 (S/N)1 so that in the saturation regime

where m and n are the number of samples taken, respectively, in time and in space per transmitted pulse, and M is the number of pulses averaged. Independent spatial samples may be obtained by use of a detector array where the elements are separated by a distance large compared with a coherence area in the detector plane; each element would, of course, require a separate processing system. As we have seen, independent temporal samples from a single output pulse have to be sep­arated in time by more than the reciprocal of the return signal bandwidth. Independent temporal samples from successive output pulses are obtained if the pulses are separated in time by more than the reciprocal of the bandwidth of the atmo­spheric return that would be obtained using a strictly mono­chromatic laser output, i.e., if the scattering particles have time to move significantly between samples; the validity of this averaging depends on the mean particle density remaining constant over the observation period.

The output ratios (S/N)1 can now be substituted in Eqs. (2) and (7). If the sampling systems for both frequencies are identical so that m, and M are the same for all four measure­ments of the return flux, we obtain

The conclusion can be drawn that, while the signal term [e.g., in Eq. (5)] is a function of range through ηh and δh and there may be signal processing advantages in minimizing this range dependence, the signal to noise ratio is independent of both range and laser energy provided the very conservative con­dition 2ηηhδh » 1 is satisfied. For a measurement involving given values of nnσρΔr, the sensitivity of a DIAL system employing heterodyne reception therefore depends only on the number of independent samples obtained, which is max­imized within a given observation time by use of a high bandwidth system. This is at variance with, for example, the implications of Fig. 2 in Ref. 2. The discussion has been of course concerned with measurements related directly to the mean return flux; it will not apply in anemometry where measurements are made of the frequency (or the time between zero crossings) of the photodetector output.

This work was supported in part by a contract with the' Commission of the European Communities research direc­torate.

3864 APPLIED OPTICS / Vol. 17, No. 24 / 15 December 1978

References 1. R. L. Byer, Opt. Quantum Electron. 7, 147 (1975). 2. T. Kobayasi and H. Inaba, Opt. Commun. 14, 119 (1975). 3. B. J. Rye, "Antenna Parameters in Incoherent Backscatter Lidar,"

accepted for publication in Appl. Opt. The derivation of Eq. (2) is parallel to that of (i2) in Appendix A in that paper, following the initial premise that the output charge from the photodetector is

The assumptions made in deriving Eq. (3) are that +ΦL, » +ΦR,, which should be valid in practical systems using ambient atmo­spheric backscatter at ranges greater than about 100 m, and that the return field is statistically stationary, i.e., that the return field correlation function gs(t1;t2) = gs(t1 – t2) = gs(); strictly this cannot be the case for a lidar system, but it is reasonable to suppose the assumption has approximate validity if the return flux varies only slowly over the sampling time d.

4. C. M. Sonnenschein and S. A. Horrigan, Appl. Opt. 10, 1600 (1971).

5. S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944); 24, 46 (1945); 27, 109 (1948).

6. W. B. Davenport, Jr., and W. L. Root, Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

7. H. Z. Cummins and H. L. Swinney, Prog. Opt. 8, 123 (1970). 8. E. Jakeman, C. J. Oliver, and E. R. Pike, Adv. Phys. 24, 349

(1975).