# Optimal bandwidth for topographical differential absorption lidar detection

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<ul><li><p>Optimal bandwidth for topogrdifferential absorption lidar de</p><p>e</p><p>2non-signal-noise currents1 such as a detector dark usually set to a few hundred hertz to avoid DC</p><p>current, an amplifier-noise current, and a back-ground-noise current. All these noise sources de-pend on the square root of the effective electronicbandwidth used for the detection. As the band-width f increases, V0 increases and so does the noiselevel. The rate of increase of the noise level and thedetected peak signal determine the maximum pos-sible SNR as a function of the bandwidth f. Theoptimal choice of bandwidth for which a maximumSNR is obtained depends on the signal temporalpulse shape and the temporal broadening of thebackscattered signal that results when the target isnot perpendicular to the incidence. For an airbornescanning lidar the angle between the incidence and</p><p>fluctuations and 1@f noise contribution. Thus thebandwidth of the amplifier is approximately itshigh-frequency cutoff f.Different low-pass filters 1e.g., Butterworth, Che-</p><p>byschev, Bessel2 can be designed3 such that thedifferences among them are the slope steepness ofthe frequency roll-off response beyond the 23-dbhigh-frequency cutoff, the flatness of the frequencyresponse within the passband bandwidth, and thetime delay for the different frequency componentscomprising the input signal. When the roll-off re-sponse beyond the 23-db-frequency cutoff is smooth,the time delay is small; as the filter response be-comes steeper for frequencies beyond the bandwidth1i.e., approaching an ideal filter2, the time delayincreases; the output signal is shifted in the timedomain, together with the distortion of the pulseshape that is due to the filters spectral attenuationof the different signal-frequency components. Inthis study, a simple single-pole RC low-pass filter,</p><p>The author is with Science and Technology Corporation, 500Edgewood Road, Suite 205, Edgewood, Maryland 21040.Received 23 May 1995; revised manuscript received 20 Septem-</p><p>ber 1995.Avishai Ben-David</p><p>A detected laser signal backscatteresponse of a high-pass filter to ancavity before a laser pulse is eamplifier. The model is used to mproper choice of the integration timthe backscattering target.Key words: Lidar, differential</p><p>Optical Society of America</p><p>1. IntroductionIn differential absorption lidar 1DIAL2 detection,when the return signal is backscattered from atopographic target, the peak value of the detectedsignal is used in the DIAL analysis and the signal-to-noise ratio 1SNRp2 is determined by the ratio of thepeak value of the detected backscattering signal 1V02to all noise sources. The backscattering signal levelis usually too low to be measured, and an amplifier isneeded to increase the return-signal level to a mea-surable quantity 1millivolts2. The main contribu-tions of noise in CO incoherent detection are from0003-6935@96@091531-06$06.00@0r 1996 Optical Society of Americaaphicaltection</p><p>red from a tilted target is modeled with a laser-pulse shape as aexponential input that describes the gain buildup within the lasermitted and a single-pole low-pass RC filter for the electronicaximize the signal-to-noise ratio of the detected peak signal with aconstant t as a function of the laser-pulse shape and the tilt angle of</p><p>absorption lidar 1DIAL2, signal-to-noise ratio, detection. r 1996</p><p>the targets normal may vary as a function of thescanning and the flight geometry. Therefore aproper choice of bandwidth and the use of a variablebandwidth may increase the SNRp of the lidarmeasurements. In this paper, the optimal band-width is given as a function of the laser-pulse shapeand the targets tilt angle to the laser incident beam.The effective bandwidth of two low-pass filters isgiven by2 f 5 f1 f2@31.051 f12 1 f2221@24, where f is the23-dB high-frequency cutoff of the two components,one of which can be the detector and the other theelectronic amplifier. The lower-frequency cutoff iscomposed of one resistor and one capacitor, is used,</p><p>20 March 1996 @ Vol. 35, No. 9 @ APPLIED OPTICS 1531</p></li><li><p>for which a simple analytical expression can beobtained and, as is shown, provides a reasonable</p><p>ductive HgCdTe detector 1time constant of 7 ns2 and aseventh-order Bessel 44-MHz bandwidth amplifierapproximation to the more complicated seventh-order Bessel amplifiers used in the lidar system.The roll-off slope beyond the 23-db cutoff frequencyis less steep for a simple RC filter 1i.e., a first-orderButterworth2 than for the seventh-order Bessel, andthus the output signal from the RC filter will belarger because high-frequency components of thesignal beyond the 23-db amplifier cutoff frequencywill be less attenuated. In addition, the time delayincreases as the order of the filter increases.</p><p>2. Laser-Pulse ShapeA gain-switch or a Q-switch laser can be viewed as aprocess for which a capacitor stores energy while apopulation inversion of excited atoms at a higherenergy level is built up in the active medium 1e.g., agas mixture He:CO2:N2 for a CO2 laser or a semicon-ductive material such as neodymium-doped yttriumaluminum garnet in a Nd:YAG laser2. When theexcited atoms return to the ground level, opticalradiation is emitted as a laser pulse. The storedenergy in the population inversion can be releasedspontaneously 1gain switch2 or by an electro-opticQ-switch device. This process can be simulated asthe response of a high-pass filter to an exponentialinput 1 2 exp12t@t12 for which the laser pulse VL1t2 ~exp12t@t22 2 exp12t@t12. The time constant t2 5 RCof the resistorcapacitor high-pass network is re-lated to the fall time of the laser pulse, and the timeconstant t1 is related to the rise time of the pulse.These time constants are a few nanoseconds in aNd:YAG laser and a few tens of nanoseconds in a CO2laser. In a CO2 gas laser, the pulse shape is charac-terized by an additional time constant t3 that resultsin general from the N2 gas and produces a long tail 1afew microseconds2, although the long pulse tail al-ways exists and can be seen in a log scale even ifthere is no nitrogen.4 Thus the falling-edgeexp12t@t22 laser-pulse shape is divided into two com-ponents, a exp12t@t22 and 11 2 a2exp12t@t32 and thelaser-pulse shape is given by</p><p>VL1t2 5 a exp12t@t22 1 11 2 a2exp12t@t32 2 exp12t@t12,</p><p>112</p><p>where t1 , t2 , t3. The peak value of VL1t2 is at t 5tL, where tL 5 t1t21t2 2 t1221ln1t2@at12; is exact for a 51; and for a . 0.8 is within a few percent of the peak.Empirical analytical forms for a CO2 laser-pulse</p><p>shape are given by Zhao et al.,4 in which the pulseshape is represented with six free parameters andequations for a short and a long time interval, and byGurdev et al.,5 who used a continuous analyticalform containing three free parameters. Both formscannot be integrated analytically, and thus it isadvantageous to use Eq. 112, which has a simpleanalytical form, can be easily integrated, and quali-tatively relates to the lasing process. An example ofthe 10R20 CO2 laser line measured with a photocon-</p><p>1532 APPLIED OPTICS @ Vol. 35, No. 9 @ 20 March 1996is given in Fig. 11a2 with a simulated laser pulse VL1t21a 5 0.91, t1 5 35 ns, t2 5 60 ns, and t3 5 800 ns2,and a simulated short pulse 1a 5 1, t1 5 3 ns, andt2 5 4 ns2 typical of a Nd:YAG laser is shown in Fig.11b2. The bandwidth of the CO2 laser pulse is ap-proximately 10 MHz.</p><p>3. Detected Pulse ShapeAll photons transmitted within a time interval Dt 52U L tan1c2@C, where L is the distance to the target,C is the speed of light, U is the incidence full-widthbeam divergence 1i.e., UL is the diameter of theilluminated spot size on a perpendicular target at adistance L2, and c is the angle between the targetnormal and the incidence light, will arrive at thedetector at the same time. This time interval Dt is2DL@C, where DL is the projected length of theilluminated spot size on the incidence direction.The broadened pulse shape Vi1t2, which is incidentupon the detectorwhen backscattered fromaLamber-tian tilted target with reflectivity r 5 r0 cos c, wherer0 is the Lambertian reflectivity for a perpendicularincidence, is given by</p><p>Vi1t2 5r</p><p>Dt e0t</p><p>VL1t82dt8 t , Dt,</p><p>Vi1t2 5r</p><p>Dt et2Dtt</p><p>VL1t82dt8 t $ Dt. 122</p><p>Fig. 1. 1a2 CO2 laser pulse 110R202 measured with a 44-MHzamplifier and a photoconductive HgCdTe detector 1time constantt 5 7 ns2 and a laser pulse VL simulated with Eq. 112 1a 5 0.91,t1 5 35 ns, t2 5 60 ns, t3 5 800 ns2. The pulse bandwidth isapproximately 10 MHz 1t 5 16 ns2. 1b2 Simulated short pulse1a 5 1, t1 5 3 ns, t2 5 4 ns2 typical of a Nd:YAG laser. The pulsebandwidth is approximately 160 MHz 1t 5 1 ns2.</p></li><li><p>As the target angle c approaches zero, the tempo-ral broadening Dt approaches zero and Vi1t2 backscat-</p><p>current 1proportional to the incident optical power inwatts2 to the overall standard deviation of the fluctua-tered from the perpendicular target converges to theincident laser-pulse shape VL1t2. The pulse shapeV01t2 detected by a receiver with a time constant t 5RC for which the rise time tr 5 2.2 t 5 0.35@f and f 51@2pt is the 23-db high-frequency cutoff backscat-tered by a Lambertian target at an incident angle c,is given by</p><p>V01t2 5 t21 e0</p><p>t</p><p>rVi1t82exp321t 2 t82@t4dt8. 132</p><p>The incident input pulse shape Vi1t2 and the outputdetected pulse shape V01t2 can be easily integratedanalytically with the laser pulse VL1t2 given by Eq. 112and are not given explicitly. The change in thedetected signal magnitude that is due to the varyingtarget distance within the illuminated spot thataffects the field of view 1~1@L22 and the atmospherictransmission are ignored, and only the detectedpulse shape V01t2 is considered.The peak value of the incident pulse Vi1t2 on the</p><p>detector in Eq. 122 can be computed within 10% forVi1t 5 ti2, where ti 5 31tL2 1 Dt221@2 1 tL 1 Dt4@2. Thepeak value of the detected pulse V01t2 in Eq. 132 isgiven within 10% by V01t 5 to2, where to 531ti2 1 t221@2 1 ti 1 t4@2 for t , 2 t2 if a 5 1 3i.e., there isno long N2 tail in the laser pulse VL1t2 in Eq. 1124 andfor t , t3 if a 1. When the integration time t islong, the peak value of V01t2 is given within 10% byV01t 5 to2, where to 5 t2t1t 2 t2221ln1t@t22 for t . 2t2and a 5 1, and at to 5 t3t1t 2 t3221ln1t@t32 for a 1and t . t3. When the temporal broadening Dt fromthe tilted target is very large, Dt : t2 for a 5 1 andDt : t3 for a 1, the input pulse Vi1t2 can beapproximated with a square pulse of width Dt, andthe peak value of the detected pulse is at t 5 Dt,V01Dt2 5 53t3 2 t1 2 a1t3 2 t224Dt21631 2 exp12Dt@t24, inwhich the first term on the right-hand side is thetotal energy in the transmitted laser pulse dividedby the temporal broadening Dt 1i.e., the average peaksignal2 and the second term is the effect of the RCintegration time constant of the amplifier.The transmitted laser pulse and the received laser</p><p>pulse 1measured with a cooled photovoltaic HgCdTedetector of 58-MHz bandwidth2 from a hard targetat an angle near perpendicular to the incidence 1i.e.,c = 02 were measured with a seventh-order Besselamplifier of 5-, 3-, and 1-MHz bandwidths. Thepeak signal was reproduced within 20% with Eqs. 122and 132. The temporal pulse shape was reproducedwell with the 5- and 3-MHz amplifier bandwidths,with the exception of a very long time delay 1hun-dreds of nanoseconds2 for the peak signal that wasobservedwith the 1-MHz seventh-order Bessel ampli-fier bandwidth and was not reproduced in the simu-lation.</p><p>4. Signal-to-Noise RatioThe SNR of the detected backscattered signal V01t2 isdefined as the ratio of the peak of the mean signaltions in the non-signal-noise current 1proportional tothe noise-equivalent optical power in watts2. WhenPoisson statistics are used, where the variance of thefluctuations is equal to the sampled average, allnon-signal-related noise sources such as dark-current noise, background-noise current, and ampli-fier-noise current contribute a noise current that isproportional to 1@t 1i.e., proportional to f 2. Thusthe SNR is proportional to V01t2t, for which themagnitude of V01t2 depends on t and on the temporalbroadening Dt that is due to the tilted target. Aspeckle noise that is important in CO2 DIAL detec-tion, is inversely proportional to the square root ofthe number of coherent speckle cells in the receiveraperture6 and is proportional to the signalmagnitude.Thus the SNR that is due to speckle is a constantthat depends on parameters such as wavelength, thebeam-divergence, receivers, aperture, and distanceto the target, but does not depend on t.The maximum value of the detected signal V01t2</p><p>increases and approaches the maximum value of theinput signal Vi1t2 as the integration time constant tdecreases. However, as the maximum value of thedetected signal increases, so does the noise increase,because a higher bandwidth is required for thedetection. As t increases, the noise decreases 1alower bandwidth is required2, and the maximumvalue ofV01t2 also decreases. Themaximum value ofthe detected peak signal SNRp ~ V01t2t is approxi-mately 1i.e., within 15%2 at t 5 t1 1 t2 1 ti and can beeasily computed from the maximum value of V01t2 inEq. 132. For a square pulse of width Dt, it can beshown that the peak output signal from a low-passfilter 1for a unit input signal2 is at t 5 Dt and of avalue 1 2 exp12Dt@t2. A matched filter with abandwidth f 5 1@2Dt and thus t 5 0.3182Dt resultsin a peak output signal of 95.7%. However, themaximum value of SNRp of the peak output signal isfound for t 5 0.7959Dt, for which the peak outputsignal is only 71.5%, resulting in the improvement ofSNRp by a factor of 1.182.SNRp 5 V01t2t, assuming a noise value of 1 at t 5</p><p>1 ns, and the detected peak signal V0 for a laser pulseof a unit peak value backscattered from a Lamber-tian target, with reflectivity r 5 1, perpendicular tothe incident laser pulse 1i.e., at c 5 02 are shown inFigs. 21a2 and 21b2, which correspond to the laserpulses shown in Figs. 11a2 and 11b2, respectively.Figures 21a2 and 21b2 show that although the peakdetected signal decreaseswith increase of the integra-tion time t, the SNRp reaches a maximum value foran optimal value of t. The improvement of SNRp forthe detected peak signal with an optimal choice ofthe integration time constant t is shown in Fig. 3 fora laser pulse of a unit peak value backscattered froma Lambertian target with reflectivity r 5 1, at adistance L 5 10 km and at an angle c. For the CO2laser pulse shown in Fig. 11a2, the beam divergence is2.8 mrad, and for the short pulse it is 0.5 mrad,</p><p>20 March 1996 @ Vol. 35, No. 9 @ APPLIED OPTICS 1533</p></li><li><p>which is typical of a Nd:YAG laser pulse 3Fig. 11b24.Figure 3 shows the temporal broadening Dt from thetilted target and the optimal time constant t tomaximize the SNRp for backscattering from thetilted target, as well as the improvement of the SNRpand the reduction of the peak value of the detectedsignal V0 when the integration time constant t isused over the SNRp and the peak detected signalwhen integration time t 5 16 ns 1i.e., a 10-MHzbandwidth, which is approximately the bandwidth ofthe CO2 laser pulse2 i...</p></li></ul>

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