# optimal bandwidth for topographical differential absorption lidar detection

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Optimal bandwidth for topogrdifferential absorption lidar de

e

2non-signal-noise currents1 such as a detector dark usually set to a few hundred hertz to avoid DC

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

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-

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,

The author is with Science and Technology Corporation, 500Edgewood Road, Suite 205, Edgewood, Maryland 21040.Received 23 May 1995; revised manuscript received 20 Septem-

ber 1995.Avishai Ben-David

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

Optical Society of America

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

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

absorption lidar 1DIAL2, signal-to-noise ratio, detection. r 1996

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,

20 March 1996 @ Vol. 35, No. 9 @ APPLIED OPTICS 1531

for which a simple analytical expression can beobtained and, as is shown, provides a reasonable

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.

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

VL1t2 5 a exp12t@t22 1 11 2 a2exp12t@t32 2 exp12t@t12,

112

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

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-

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.

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

Vi1t2 5r

Dt e0t

VL1t82dt8 t , Dt,

Vi1t2 5r

Dt et2Dtt

VL1t82dt8 t $ Dt. 122

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.

As the target angle c approaches zero, the tempo-ral broadening Dt approaches zero and Vi1t2 backscat-

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

V01t2 5 t21 e0

t

rVi1t82exp321t 2 t82@t4dt8. 132

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

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 th

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