Ramachandra Reddy, G

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Indian J. Phy*. 52B. 214-217 (1978)Letters to the EditorOn the power spectrum of speckles in coherent imagingG. TtAMArHANDBA REDDYDepartment of Phy>ncs, Regional Engineering Colle^ ge Wamw<7aZ-50B004 (A .I\)(Received 30 November 1976)Tho speckle pattern in the image is generally reduced by moving the diffuser rapidly. Thin is, however, not possible in all cases viy. in holographic imagery. To minimise the speckle in holographic imagery a method (Dainty ik Welford 1971) which is basically a spatial frequency technique was suggested to have a moving aperture in the pupil plane o f the imaging lens. Recently Hariharan Hegedus (1974) have obtained an expression for the power spectrum of >speckles in the image o f a coherently illumiiiattKl diffusing object in terms o f the shift o f the sampling aperture in the spatial frequency plane between successive apertures by extending the same procedure to the case o f a moving random mask. They have assum(Mi the diffuser to be stationary. In the prevsont communication we would like to show the effect o f a moving diffuser associated, with a stationary diffuser on the power spectrum o f speckles.r11 1. r v - l1 JFig. 1. X>2 are diffuser, Dx is stationary, Dg is mlving. Li, lenses. H—Aperture.Let us consider the coherent optical system as shown in the figure the symbols and notation used therein being the same as those used by Hariharan S& Hegedus (1974) for convenience. The system gives the image pf an object plane consisting of twp diffusers with complex amplitude tranamittanoes djj(r) and <?2(^ ) respectively where r is a point specified by the co-ordinates (a?, y). Let assume that the diffuser with transmittance d (^r) is moving while the otherm^peefdes in coherent imaging 216is stationary. Tho omorgont Jigh.t from difftist r^s is modviatmi by iv factor t{r) and tho object function can be written as/(r, vt) — t{r)d^(r)dt{r—vt) (1)where v is the constant velocity o f the juov^g diffuser, ff h{r) is the coh.rteiit system impulse then under t h e  iissmnptiui^ o f  unit magnilication tlio image amplitude can be written asg{r, vt) =- f(r, ot)\k{r)where the symbol * denotes convohition.(2)Now for n equal exposures with a shift ofth.e ajKM tu.re in betutsm two sucg»»s- sive ones the resultant illuininaiu (» /(r) at a|iy point in the imag(‘ i)lane will be the sum o f the individual illuminances and be written asHr) =  S  Ii(r, vt) (3)where Ii{r,vt) - [ gr/(''. ct) P  and f/dr, r ) f{r. vt)*ht{r), ht{r) being the system impulse for the i-th exposuns The auto-correlation of th.e resultant  ^ illuminance can be written ash’ //(r,. /'a, vt) S <  /<('•,, vt)> <  H):i j-i l/t /{ (r ,,r ,,r t) I i,i i }wTile a v e r a g e  p o w e r  s p e c t r u m  o f  t h e  i m a g e  i l l c m i n a n c t *  c a n  b e . e x p r i 'S S t s l  a s  t h e  sum o f two terms as follows.«2  vt) -= »<) +  i 22(M,. « 2, vt) (3)W'here vt) is tlie double Fourier transform of the illuminance*r^ , vt). In other words as we can wriU* fit and as the resjMictive. Fourier transforms of the two teims of the ('xpression (4). It is shown by Ijowenthal & Joyeux (1971) that the term < I (r ,r t )>  in th<‘ image does not <lep!n*l either on position r or on time t. Th,e average pow'ei sjM>ctnim, replacing and « j, is given (Hariharan & Ht*gedus 1974) a.sil^(n, u ,v l)  - \ ^ {u )M u )f  ... (6)whore ^ u )  is the Fourier transform of the irradiance o f tlie object and J^u) is the incoherent transfer function ol tl\e system. The above equation therefore gives the powder spectrum of a spatially incoherent object having the same irra­diance I t{r) j * as the object transitarency.The second term Wg, vt.) is jimre important for us as it gives the powerspectrum o f the speckle. For two linear systems with inputs fi{r, vt), f}{r, ®(),216 G. Biamachandra Reddyimpulse responses hi(r), hj(r) and outputs gt(r, vt), vt) respectively, the cross- correlation o f the outputs> can oxproHScc* in terms o f th(  ^ cross correlation o f the inputs as (Papoulis 1965).„ (r,, t\, vt) K {»■,. /-j, t)* hjir^)* /t<(r/l) . . .  (6)where the^  subscript r is the variable on whu;b, tl>i‘ operation is carried out, keeping the other variables constant. ThusU.y^  ot) < > ^  j /’.J, vt)^ ^   ^ (^ *2) I1 2... (7)wherti R (/*j, >0, vt) is the auto correlation innetion o f the obj(Sit amplitude /(r , vt).I f  the doubles Pcnirier transform of Hff isr('Wi, vt) ^ ’^2> ’^0thenr.y. vt) - S I V(u ,^ vt)Hi(v )^H* (^ w.)lU... (9)In order to obtain thc» aveiago power siHxitrum o f the speckle* v\e l\avc to replaces and u.y bv ti. Then r(w, a, vt) becomes S/f(v, vt) and lieneer(? ,^ u, vt) - -  Sff(u, vt) = - vt). ... (10)Hence the average pow(vr S2>ectrum of the speckle is given byLUu. ti. vt) ^  I Eff(iK vt) 1 S  \Hi(u)H^*iu)M Hi*(^u)Hj(--v)] ... (11)The aboV(‘ (^xpnvdsion clearly sho>\s that the poner spectnim o f speckle depends on the jnoving diffuser. In the absence o f the moving diffuser ./ /^/(O, vt) becomes Eff(0 ) which is constant and the expres.sion (11) is then identical to that dei’ived by Hariharan «Sr Hogedus (1974).The illustiatcv tlie aboves result, let us coasider a rectangular aj[H5rtiU’e o f unit widthrect('«^ ) ^-1 | /^ j <  1/2o \ u \  >  1/2 ... (1 2 )which is moving in tile Fourier plane with unit velocity along a line patallel to its width diu*ing the exposure time T, so that the total distance U through whicli the aperture moves is T  — U which is assuniod to be greater than its widtli.Speckles in cohermt imaging 217The displacements of the aperture at any two instants and during exposure are given by(u) =  rect(t«—tti) (u) =  rect(iA—ita) (13)Uq being the initial positions of the aperture. Now proowding on tlie same lines as Hariharan & Hegedus (1974) we got |lie contribution to the j^ower spec­trum o f the speckle along the direction o f t ie  movement from the aperture at those two positions as ?=  I 1 ^[rect(#—'ai)roct(ii—Wg)]* [r ect (i%— w)rect (^2 ) I •Integrating over all values of and(14)u uu uwhere u ,^ U2 and u are always positive and Wg <  1—From eq. (15) it is clear that the power spectrum of the speckle is always weighted by the moving diffuser an<l tlie same logic can be extended tf) the case o f moving random mask.AcknowledgmentThe author is grateful to Dr. P. K. Mondal foi bis giutlancu.R efebencesDainty J. C. & Welford W. T. 1971 Opt. Commun., 3, 289.Hariharan P. & Hegedus Z. S. 1974 Opt. Act. 21, 683.Lowenthal S. & Joyeux D. 1971 J. Opt. Soc. Am. 61, 847.Hariharan P. & Hegedus Z. S. 1974 Opt. Act. 21, 345.PapouliB A. 1965 Probability, random variables and Stochastic Processes, McGraw-Hill, N w   ^ York*B-12

On the Power Spectrum of Speckles in Coherent Imaging

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On the Power Spectrum of Speckles in Coherent Imaging

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