Position Error Estimation of a Laser Illuminated Object

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1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 3, 1 (2011, Position Eo Estimation of a Lase Illuminated Object Ž. P. Babaić Abstact: Position eo a lase illuminated object (taget is analyzed. The elations fo the displacement signal and pobability density function of displacement signal by lase illuminated object ae given. The eo of displacement signal and the position eo ae deived. The eo of displacement signal of lase illuminated object vesus to signal-to-noise atio (SNR and the mean value displacement signal. The eo of displacement signal is invese popotional to the oot squae signal-to-noise atio, and inceases with the mean value displacement signal. The minimum eo of displacement signal is oot squae two times smalle than maximum value, fo constant signal-to-noise atio. The position eo is popotional to the eo of displacement signal.the minimum position eo is obtained when the cente of the spot is in the cente of quadant photodiode, and inceases with x 0 /, fo constant SNR. The position eo apidly inceases at the limits of the measuement ange x 0 / = ±1. Keywods: quadant photodiode, displacement signal, pobability density function. 1 Intoduction Thee ae a numbe applications in which accuate positioning ae needed, as industy and amy [1]. That applications main chaacteistic is accuacy positioning of an object. Lases systems positioning have special teatment, because those esolution is bigge than ada and othe. A numbe applications lases positioning include tacking illuminated taget and measuement its angula position [2], [3], estimation vibation effect of Satellite [4], and measuement multidimensional displacement in space [5]. These and othe applications can be used position sensitive detectos with quadant photodiode (QPD [6] o lateal effect photodiode (LEP [7] to measue lateal displacement in two pependicula plane. Highesolution multidimensional displacement monitoing system had development with fou quadant photodiode [5] that can be used monito six degees feedom. They have pesented esults that lateal esolution bette than 50nm and angula displacement bette than 0.25 mico-adians [5]. Theoetical displacement signal analysis had disused in [1], [2], [3], [4]. In papes is pesented that main paamete is signal-to-noise atio, which limited accuacy of positioning in lase systems with quadant photodiode. Also, has found that the position eo is Manuscipt eceived Novembe ; accepted Febuay Ž. P. Babaić is with the State Univesity of Novi Paza. 59

60 Ž. P. Babaić changed with mean value of displacement signal [8]. The expession fo pobability density function of displacement signal fo Gaussian noise distibution is pesented in [9].

2 60 Ž. P. Babaić changed with mean value of displacement signal [8]. The expession fo pobability density function of displacement signal fo Gaussian noise distibution is pesented in [9]. In this pape a new appoach to estimate the position eo of lase illuminated object by quadant photodiode is given. This appoach is statistically, whee the position eo is calculated on base the eo of displacement signal, which obtained fom pobability density function of displacement signal. 2 Displacement signal The positioning lase illuminated object based on focusing of eflected fom object lase beam by eceiving optics. Displacement angle between the eceive optical axes and eceiving iadiance is shown on Figue 1. The pupose of the optical eceive is to collect the eflected optical enegy fom the illuminated object. Fig. 1. Lase beam and eceive of eflected lase beam The plan convex lens, which is placed at the input of the optical eceive, collects the incoming optical enegy on the quadant photodiode (QPD. Figue 2 shows base optical geomety two main components a thin lens, and a quadant photodiode. The incident eflected lase enegy incomes on the thin lens and its focuses on the quadant photodiode suface. To minimize the eo the plan convex a spheical lens is mostly used, because of its minimal spheical abeation coefficient. The QPD is located behind the lens at the distance d fom it. Figue 2 gives the example of QPD located in font of focal plane, e.g. d < f, whee f is the focal length of the lens. As the QPD isn t placed at the focal plane, a light spot with an appoximately unifom distibution of iadiance is fomed on its suface. In Figue 2 incident adiation incomes with angle δ espect to nomal on the suface of lens and poduces spot with cente (C on the quadant photodiode. Catesian coodinate system sets in cente of photodiode, and the cente of spot C has coodinates x 0,y 0. Figue 2 shows

3 Position Eo Estimation of a Lase Illuminated Object 61 that point C is function of incident angle δ, fo constant distance (d between thin lens and photodiode. Fom geomety in Figue 2 can be obtain tgδ h = x 0 /d i tgδ v = y 0 /d,whee δ h and δ v ae the angles in hoizontal and vetical planes, espectively. Fig. 2. Geomety between thin lens and quadant photodiode A spot position on quadant photodiode diamete2a shows in Figue 3. The diamete of spot 2 is function of distance d and focal length f of thin lens. Fig. 3. A spot position on quadant photodiode suface The displacement signal in nomalized fom, fo hoizontal and vetical plane, fom Figue 3, ae: ε x = (i 1 + i 4 (i 2 + i 3 (i 1 + i 4 + i 2 + i 3 = i x i ε y = (i 1 + i 2 (i 3 + i 4 (i 1 + i 2 + i 3 + i 4 = i y i (1 whee i i is the cuent i-th quadants (i =1,2,3,4. The cuent i-th quadants is i i = RpP 0,whee P 0 is the total eceiving flux, q is pat of one (1 p 0, a R is the esponsivity of photodiode. The total cuent is the sum of fou cuents (i Σ = RP 0.

4 62 Ž. P. Babaić Thee ae two models fo calculation displacement signal as function displacement the cente of the spot espect to the cente of the photodiode. The fist models that assume constant distibution of the iadiance on the sensitive suface photodiode and limited sizes of the spot with cicle shaps. Othe models assume Gaussian o sinc distibution of the iadiance on the sensitive photodiode suface. The displacement signal, fo constant iadiance distibution on the cicle shap of the spot, fom Figue 3, ae: ε x = 2 x 0 1 x2 0 π 2 + acsin(x 0, x 0 ε y = 2 π x 0 1 y acsin(y 0, y 0 (2 whee ae x 0 = dtgδ h, y 0 = dtgδ v. Cente of spot is appoximately popotional to incident angle, because tgδ δ fo small value of δ (δ The atio d/ is the constant of the position senso (d/ = K D. The displacement signal, fo Gaussian distibution of the iadiance with cente (x 0,y 0 and squae shap of photodiode size2a, ae: ( ( ( 2e f 2σ x0 + e f a x0 2σ e f a+x0 2σ ε x = ( ( e f a x0 2σ + e f a+x0 2σ ( ( ( 2e f 2σ y0 + e f a y0 2σ e f a+y0 2σ ε y = e f ( a y0 2σ + e f ( a+y0 2σ (3 whee ae: σ- is the standad deviation of two-dimensional Gaussian distibution iadiance on the sensitive photodiode suface, and ef(x is the eo function, whee e f (x = 2 π x 0 exp( u 2 du. The standad deviation σ in (3 and the adius of spot in (2 is in elationship. The pat of flux in the cicle adius espect to total incident flux is P = 1 P T 2πσ 2 2π 0 dφ 0 exp( ρ2 2 ρdρ = 1 exp( 2σ 2 2σ 2 (4 Fo example, fom (4, atio P/P T =90% of total flux lies in cicle of adius =2.1456σ. The diagams of the displacement signal fom (2, (3 ae given on Figue 4, fo =2.1456σ, (= 2.25 mm, σ =1.048 mm. The best ageement displacement signal fo Gaussian and cicula spot on the sensitive suface QPD is found aound x=0 and x/=±1, as shown on Figue 4.

Position Eo Estimation of a Lase Illuminated Object 63 Fig. 4. Displacement signal ε x fo constant iadiance and Gaussian distibution iadiance on the suface photodiode, fo =2.1456σ, (= 2.25 mm, σ =1.

5 Position Eo Estimation of a Lase Illuminated Object 63 Fig. 4. Displacement signal ε x fo constant iadiance and Gaussian distibution iadiance on the suface photodiode, fo =2.1456σ, (= 2.25 mm, σ =1.048 mm 3 Eo of displacement signal and position eo Expessions fo the displacement signal (3 and (4 ae deived with assumptions fo the spot shap geomety and the iadiance distibution. Displacement signal in eal is sum of signal and noise. The noise add to signal influence to changing cente of spot, which play aound x 0, y 0. The position eo can be undestood as the changing of the cente of spot, which can not be detected. Those changing of the cente of spot x 0 and y 0 can be estimate fom expessions fo the displacement signal. The eo of the displacement signal fom (1 fo one axis is ε x = i x i Σ (5 whee i Σ and ae constant. We assumed that the cuent diffeence is equal the standad deviation of the total noise cuent ( i x =σ t. Now fom (5 the eo of displacement signal ε x fo x-axis is ε x = 1 SNR (6 whee SNR is the signal-to-noise atio in the channel sum. Equations (6 shows that the eo displacement signal of lase illuminated object is invese popotional to oot squae of signal-to-noise atio, in channel sum. The eo of displacement signal obtained fom the deivative of the displacement signal (2, fo x-axis is ε x = 4 x 0 1 (x 0 / π 2 (7 whee x is the position eo.

6 64 Ž. P. Babaić Equation (7 shows that the position eo ( x 0 is popotional to the eo of displacement signal ( ε x. The displacement signal ε x and ε y in (1 ae combination cuent of incident iadiance and noise of photodiode. The displacement signal (1, fo one axes can be witten as: ε = S D + N 1 = u v S S + N 2 u + v (8 whee S D is the pai wise diffeence signal fo a given axis, S S is the sum signal all (fou quadants, N 1 is the oot mean squae (ms noise associated diffeence signal, N 2 is the ms noise of the sum signal. u, v- ae the signal plus noise of sum two quadants, espectively. We assume that each quadant geneate noise with Gaussian distibution with zeo mean value and vaiance σ 2 n. Then u and v fom (8 epesents signal fom pai quadants of photodiode u = ū + N u v = v + N v (9 whee ūand v ae mean value, and N u and N v ae fluctuation ofuand v, espectively. Pobability density function (pdf of displacement signal is obtained, on base known theoy fo jointly nomal andom vaiables u and v, pobability density functions f (u, vand tansfomation its in f(ε [10], in the fom ( 1 ρ 2 ū2 f (ε = π(1 + ε 2 2ρε exp + v 2 ρ(ū 2 v 2 [ ( π 2σp(1 2 ρ B e f ( B ] exp B2 ( whee ae: ρ is the coelation between N 1 = N u N v and N 2 = N u + N v, σ 2 p=2σ 2 n, and Bdefines as B = u(1 + ε + v(1 ε ρ ( u(1 + ε + v(ε 1 2σp (1 ρ 2 (1 + ε 2 2ρε Analysis of pdf has given in [11] whee it was shown that the coelation coefficient change both the maximum and width of pdf. Maximum width and minimum amplitude of pdf was obtained fo coelation coefficient equal zeo. That means the wost case fo the eo of displacement signal is when uncoelated noise between pai quadants (ρ=0. Useful way to analyze pdf of displacement signal f(ε that the mean value pai (u,v substitute with mean value of the displacement signal ( ε.the mean value of displacement signal fom (8 and (9 is: ε = ū v (11 ū + v

Position Eo Estimation of a Lase Illuminated Object 65 Signal-to-noise atio in the channel sum is SNR = (ū + v2 4σ 2 n (12 Afte substitute (11 and (12 in (10 and aangement equation fo B fo ρ=0

7 Position Eo Estimation of a Lase Illuminated Object 65 Signal-to-noise atio in the channel sum is SNR = (ū + v2 4σ 2 n (12 Afte substitute (11 and (12 in (10 and aangement equation fo B fo ρ=0 becomes B = ū(1 + ε + v(1 ε 2σp 1 + ε 2 = 1 + ε ε 1 + ε 2 SNR (13 Used to (11 and (12 can be witten ū 2 + v 2 4σ 2 n = 1 2 SNR(1 + ε2 (14 Pobability density function (10 fo uncoelated noise (ρ=0 as function the mean value of displacement signal is given on Fige 5. Fig. 5. Pobability density function of displacement signal as function fo ε=0.0, 0.5 and 1, and SNR=10 Fom Figue 5 can be seen changes both the maximum and width of pdf. Maximum width and minimum amplitude of pdf was obtained fo ε=1. Pobability density function of displacement signal fom (10, fo ρ=0, afte applied (13 and (14 and appoximation fo the complementay eo function efc(x exp(-x 2 /(xπ 1/2, becomes f (ε SNR 2π 1 + ε ε (1 + ε 2 exp 3/2 ( SNR(ε ε2 2(1 + ε 2 Pobability density function (15 is good appoximation pfd (10 fo uncoelated noise. The eo of this appoximation is smalle than 0.8% fo the wost case: ε=0, and small SNR=5, (15

8 66 Ž. P. Babaić Fig. 6. Appoximation eo fo SNR=10 and ε =0. and descies vey fast with inceases SNR. In Figue 6 is shown appoximation eo vesus displacement signal ε, fo mean value ε and signal-to-noise SNR as paametes. Fom appoximation fom pdf of displacement signal (15 can be deived the eo of displacement signal. The eo of displacement signal pesented small ange ε aound the mean value ε in which pobability is equied. This ange ε usually calculates fo pobability 50% (CEP-the Cicula Eo Pobability. Pobability positioning is given by P p = ε+ ε ε ε f (εdε (16 whee ε is the eo of displacement signal. Integal (16 can will solved in closed fom fo pobability density function is given (15. Afte substitution (ε ε2 = x integal fom (16 in the unlimited fom becomes 1+ε 2 SNR f (εdε = 2π 1 2 x exp ( SNR 2 x dx = 1 2 e f ( SNR 2 ε ε 1 + ε 2 Fom (16, and (17 pobability positioning that the displacement signal ε lies in ange ε aound ε becomes [ ( ( ] P p = 1 SNR ε SNR ε e f + e f ( ( ε + ε ( ε ε 2 It can be shown, that fo SNR 10, the aguments of the eo functions allow futhe appoximation of P p fom (3. 14 leading to ( SNR ε P p e f ( ε 2 + ε 2 (17

9 Position Eo Estimation of a Lase Illuminated Object 67 The elative eo of this appoximation, is the wost case ε=1, is less than 2,5%. By solving (19 fo ε we obtain the eo of displacement signal is 1 + ε 2 ε = (20 (SNR/2C 2 1 whee C =efinv(p p. As can be seen fom (20, the maximum value of ε, fo any given signal-to-noise atio is 2 times lage fo ε=1 in compaison to the minimum value fo ε=0. The minimum value of ε is invesely popotional to SNR, fo SNR >> 2C 2, as obtained in (2. The eo of displacement signal (20, fo SNR >> 2C 2 becomes 2C ε 1 + ε 2 (21 SNR whee an appoximation eo is smalle than In Figue 7 is shown the eo of displacement signal as function of the mean value of the displacement signal, and SNR as paamete. Fig. 7. The eo of displacement signal as function mean value, fo Pp=50% and SNR=10 db, 20 db, and 30 db. The eo of displacement signal changes with SNR and the mean value of the displacement signal as shows in Figue 7. The eo of displacement signal always is minimum fo the mean value displacement signal equal zeo and maximum fo the mean value equal plus o minus one. An example, the eo of displacement signal fom (21 and pobability positioning 50% (CEP, lies in ange between minimum value ε=0.675/ SNR, fo ε=0, and maximum value ε=0,954/ SNR, ε = ±1. Now we have two esults fo the eo of displacement signal. The fist esult (6 is obtained diectly fom the displacement signal, and second esult (21 is deived fom pobability density function of the displacement signal. Both esults show that the eo of

10 68 Ž. P. Babaić displacement signal a lase illuminated taget is invese popotional to oot squae signalto-noise atio. The eo of displacement signal given in (6 emained constant fo given SNR, but the eo displacement signal (21 changes depend on the mean value of displacement signal. The eo of displacement signal ε and the position eo x ae popotional as shows in (3. The nomalized position eo in the x diection, fom (3 is x 0 = π 4 ε 1 x2 0 2 (22 whee ε is the eo of displacement signal (21. Nomalized position eo calculated fom (22 and (21 as function nomalized position x 0 /, is shown in Figue 8. Fig. 8. Nomalized position eo as function nomalized position fo P p =0.5, and SNR=10, 100 and100 Fom Figue 8, may be seen that the nomalized position eo depends on SNR and nomalized position x 0 /. The minimum position eo is obtained when the cente of the spot is in the cente of quadant photodiode, and inceases with x 0 /, fo constant SNR. The position eo apidly inceases at the limits of the measuement ange x 0 /=±1, as shows Figue 8. 4 Conclusion The eo displacement signal inveses popotional to the oot squae of signal-to-noise atio. Fo the mean value of displacement signal equal zeo and pobability 50% the eo of displacement signal is 0.675/ SNR. The eo of displacement small incease with gows the mean value of displacement signal. The maximum value of the eo of displacement

11 Position Eo Estimation of a Lase Illuminated Object 69 signal is 0.954/ SNR, on the ends of ange positioning. Maximum value of the eo of displacement signal is oot squae times bigge than minimum value, fo constant both pobability and signal-to-noise atio. The position eo depends on SNR and nomalized position x 0 /. The minimum position eo is obtained when the cente of the spot is in the cente of quadant photodiode, and inceases with x 0, fo constant SNR. The position eo apidly inceases at the limits of the measuement ange x 0 /=±1. Acknowledgement The Ministy of Science and Technological Development of the Republic of Sebia suppoted this wok unde contact TR Refeences [1] A. MÄKYNEN, Position-sensitive devices and senso systems fo optical tacking and displacement sensing applications, Ph.D. Thesis, Depatment of electical engineeing, Oulu, [2] H. ANDERSSON, Position Sensitive Detectos - Device Technology and Applications in Spectoscopy, Ph.D Thesis, Sundsvall, Sweden, [3] M. ALKERYD,Evaluation of Position Sensing Techniques fo an Unmanned Aeial Vehicle, Ph.D. Thesis, Depatment of Electical Engineeing, Linköping Univesity, Linköping- Sweden, [4] A. S., KOPEIKA, Lase Satellite Communication Netwok- Vibation Effect and Possible Solutions, Poceedings of the IEEE, Vol. 85, 10 (1997. [5] L. NEVILLE, Y. CAI, A. JONEJA, High-esolution multidimensional displacement monitoing system, Optical Engineeing, Vol.36, 8 (1997, [6] DL100-7PCBA3, Dual Axis Psd Sum And Diffeence Amplifie Specification, 5700 Cosa Avenue, 105 Westlake Village, CA [7] QP50-6SD2, Dual Axis Psd Sum And Diffeence Amplifie Specification, 5700 Cosa Avenue, #105 Westlake Village, CA [8] W. L. WOLFE, G. J. ZISSIS, The Infaed Handbook, Chapte 22, Ann Abo, Michigan: Envionmental Reseach Institute of Michigan, [9] R. G. KAZOVSKY, Theoy of tacking accuacy of lase systems, Optical Engineeing, Vol.22,3 (1983, [10] A. PAPOULIS, Pobability Random Vaiables, and Stochastic Pocesses, McGow Hill, [11] Ž. BARBARIĆ, Position Eo of a Lase Illuminated Object, Scientific Technical Review, Vol. LII, No.5-6, (2002,

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