Observations of light backscattered from air bubbles in a viscous liquid demonstrate an enhancement due to axial focusing. A physical-optics approximation for the cross-polarized scattering correctly describes the spacing of regular features observed. The non-cross polarized scattering is not adequately described by a single class of rays

Langley, Dean S.

Marston, Philip L.

English

College of Saint Benedict and Saint John’s University: DigitalCommons@CSB/SJU

College of Saint Benedict and Saint John's University DigitalCommons@CSB/SJU Physics Faculty Publications Physics 9-28-1981 Glory in optical backscattering from air bubbles Dean S. Langley College of Saint Benedict/Saint John's University, dlangley@csbsju.edu Philip L. Marston Follow this and additional works at: https://digitalcommons.csbsju.edu/physics_pubs  Part of the Physics Commons Recommended Citation Langley DS, Marston PL. 1981. Glory in optical backscattering from air bubbles. Physical Review Letters 47(13): 913-916. https://doi.org/10.1103/PhysRevLett.47.913 ©1981 The American Physical Society VOLUME 47, NUMBER 13 PHYSICAL REVIEW LETTERS 28 SEPTEMBER 1981single-mode optical fibers with positive GVD. Byusing a precise cross-correlation scheme, wemeasured the output pulse shape from the fiberwith 2-psec precision and thereby tested both thepredictions and the validity of the nonlinearSchrodinger equation.We would like to acknowledge the careful read-ings of this manuscript and helpful comments ofM. M. T. I oy, Aa. S. Sudbo, and J. J. Wynne.This work was partially supported by the U. S.Office of Naval Research.Permanent address: Department of Physics, Fac-ulty of Science, Kyoto University, Kyoto 606, Japan.'H, . H. Stolen and C. Tkn, Phys. Rev. A 17, 1448(1978).L. F. Mollenauer, R. H. Stolen, and J. P. Gordon,Phys. Rev. Lett. 45, 1095 (1980}.36. B. Whitham, Linear and Nonlinear ~aves (Wiley,New York, 1974).Solitons in Action, edited by K. Lonrgren andA. Scott (Academic, New York, 1978).5V. E. Zakarov and A. B. Shabat, Zh. Eksp. Teor.Fiz. 61, 118 (1972) [Sov. Phys. JETP 34, 62 (1972)l,and 64, 1627 (1973) [37, 823 (1973)].R. H. Stolen and A. Ashkin, Appl. Phys. Lett. 22,294 (1973).A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23,142, 171 (1973).J. Satsuma and N. Yajima, Prog. Theor. Phys. ,Suppl. 55, 284 (1974).R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46,4921 (1975).' M. Jain and N. Tzoar, J. Appl. Phys. 49, 4649(1978), and Opt. Lett. 3, 202 (19783.'B. Bendow, P. D. Gianino, N. Tzoar, and J. Jain,J. Opt. Soc. Am. 70, 639 (1980).' B. Bendow and P. D. Gianino, Opt. Lett. 4, 164(1979}.R. H. Stolen, Proc. IEEE 68, 1232 (1980).B. Crosignani, C. H. Papas, and P. DiPorto, Opt.Lett. 6, 61 (1981).~5N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).' H. Nakatsuka and D. Grischkowsky, Opt. Lett. 6, 13(1981).The constant o.'- 1 adapts these equations to describepropagation in single-mode fibers (Refs. 6, 7, 10, ll,and 14).' The general trend of both the data and the calcula-tions as functions of increasing input power is thefollowing. For input power levels below approximately0.1 %, the pulse is undistorted by passage through the70-m fiber. As the power is increased above thisvalue, the pulse broadening, chirping, and self-steep-ening increase monotonically up to our maximum avail-able input power of 10 W.Glory in Optical Backscattering from Air BubblesDean S. Langley and Philip L. MarstonDePa&ment of Physics, Washington State University, Pullman, Washington 99164(Received 23 December 1980)Observations of light backscattered from air bubbles in a viscous liquid demonstratean enhancement due to axial focusing. A physical-optics approximation for the cross-polarized scattering correctly describes the spacing of regular features observed. Thenon-cross polarized scattering is not adequately described by a single class of rays.PACS numbers: 42.20.Gg, 42.10.Hc, 92.10.PtThe Mie solution' for electromagnetic scatter-ing by a sphere frequently does not lead to directinterpretation of the angular scattering pattern.Consequently, models have been developed to fa-cilitate an understanding of the structure in thescattered intensity present where intensity isplotted as a function of the scattering angle p orthe size parameter x =ha (h is the wave number;a is the sphere radius). These models have em-phasized the angular regions where diffraction isimportant for a drop of water in air: the rain-bow,"y = 180,' ' and p = 0 ." In the scatteringof light by a spherical air bubble in a liquid or inglass, the real part of the refractive index of thesphere is less than that of the surroundings andthe models must be significantly modified. Newphenomena appear, such as diffraction" in theregion of the critical scattering angle p, . Herewe report the first detailed observations of back-seattering by air bubbles in liquids and give amodel which describes some of the observed fea-tures. We refer to this as glory because, as inthe case of drops, ' ' the p= 180 scattering is en-hanced when x is large.1981 The American Physical Society 913VOLUME 47y NUMBER 13 PHYSICAL REVIEW LETTERS 28 SEPTEMBER 1981Van de Hulst" gave a partial explanation of theenhancement for drops by noting the axial focusingof those backscattered rays which have a nonzeroimpact parameter. When modeling this focusingin the far field, diffraction provides an essentialcorrection to ray optics because the factor in thescattered intensity which accounts for geometri-cal divergence of the rays goes to ~ as p- 180'.Examination of this factor in ray-optics modelsof scattering by bubbles' shows that this infinityis not restricted to drops. We have modeled thebackscattering with a physical-optics approxima-tion. The procedure is to (a) compute amplitudesin an exit plane in contact with the bubble via rayoptics, and (b) allow this wave to diffract to thefar field where the distance from the bubble'scenter R» ka .Figure 1 illustrates several rays which lead tobackscattering. The paths are determined by thenumber of chords p and m =m;/rn„where the re-fractive indices of the inner and outer media, m;and m„are taken to be real. Figure 1 is drawnwith rn ' =1.403 which corresponds to an air bub-ble in the dimethyl-siloxane-polymer liquid usedin the experiment. All rays satisfy sin19 =m sinp.For p =180, the off-axis (or glory) rays have 8=8 and p =p, where" 8 = pp +(2g+2- p)90', g is anonnegative integer (g =0 for rays in Fig. 1), andm & 1 requires that p - 3. The exit plane (dashedline in Fig. 1) touches C with its normal parallelto the propagation direction of the incident wave.Our description of the field in the exit plane isfacilitated by considering the propagation of awavelet de which lies close to the backscatteredpath. Figure 1 shows de for p =3; it emerges ascurve 4'e'. This curve appears to come from aringlike source at E known as the focal circle inthe analogous P =2 scattering from drops4 withW2&m & 2. The source is ringlike because thefigure may be rotated around the CC' axis. Theradius of the ring is b =a sin8. After the incidentray crosses the dashed vertical plane (the en-trance plane), the propagation phase delay forreaching the exit plane is q =ka[l —cos8 +(1—cosp)sec(8 —p) +2m p cosp]. The ray crossesthe exit plane at a radius s from C' with s/a = sinp—(1 —cosP)tan(8 tI) -Th. e radius n of arc d'e'follows from the curvature at s =b: n =k(d2q/ds') ' =a[1+~(p~ —1) 'cos8], where ~ =tanp/tan19. The spreading of the wavelet is character-ized by q = lim(d'e '/de) as de - 0 where the bar de-notes the are length. An equivalent expressionfor q is ~lim[b —s(8)]/(b —a sin8)~ as 8- 8; itsvalue from L'Hospital's rule is o./(a —a). Vectorse, (l =1,2) denote orthogonal basis vectors in boththe entrance and the exit planes; e, is chosenparallel to the polarization of the incident wave'selectric field E, exp(- iet).In the exit plane, the field E~'e, of the outgoingpth glory wave is computed by applying Van deHulst's method of first decomposing the fieldsperpendicular and parallel to the scatteringplane. '4 Exit-plane polar coordinates centeredon C' are (s, g), where g is the angle relative toe, and s and $ denote local basis vectors. We as-sume x» 1 and use Fresnel's coefficients ~, forthe internal reflections where j =1,2 for fieldsparallel to ( and s, respectively. If ~s b~ «a, -the multiple internal reflections givep=4E,' =E,q '"F' exp[iq+ik(s —b)'/2n],0, 2IIIFFIG. 1. Rays which contribute to backscatterirz.The local angle of incidence is 0 and ~ is the bubble scenter.where q =p+ g(p =p), E'(g) =c, sin'g+c, cos'g,E2(g) =~(c2 —c,) sin2$, and c,. =(-1) ~' '~r, ~ '(1r, '). The-new phase term p accounts for thecrossing of caustics or "focal lines"; its valueis'" -m(p+g)/2. The r, are evaluated at 8:r, =sin(8 —P)/sin(8+P) and r, =tan(8-p)/tan(8+p). The sign factor in c,. accounts for a geo-metrical inversion (present when j =2 and p isodd) which is not evident in descriptions of p =2glory in drops. 'The field E~' at a distant point Q is computed asfollows. The left extension of the CC' axis makesan angle y with C'Q. When y is small and C'Q=R'»ka', scalar diffraction theory and the Fraun-91428 SEPTEMBER 1981VOLUME 47, +UMBER lghof er approximation" giveP H YS ICAL RK V I K%' LETTER Sreflections (e.g. , p =0 and 2 in Fig. 1) which areproportional to a'. The strongest reflection hasp =0 and l =1; for y =0, Ioi =I@(m —1)~/(m +1)while I,' =0 ~ Since f~, does not depend on a, Ip'~ka3 and glory terms dominate the backscatteringwhen a is large.Consider a bubble with x =4000 and m =1.403 '.The strongest glory terms have g =0 and p =3, 4,and 5; the I~'/I~ for y =0 are, respectively, 1.03,0.43, and 0.16. The I~' decrease with increasingp as a result of the partial reflections in the bub-ble. The strongest axial ray gives I,'/I„=0.028.The interference of the fields depends on a andour Mie computations verify that the backscat-tered intensity is not simply proportional to aeven for this large value of x. The l =2 (cross-polarized) scattering is, however, nearly domi-nated by the p =3 glory term. Because of sym-metry, / =2 scattering vanishes as y - 0. TheI~'(y&0, $) have maxima at g =+45' and+135' andthey vanish at $ =0', + 90', and 180'. Let y =y~locate the first maxima of I~'(y, $ =45'). Thelargest l =2 terms have I~2(y~, ( =45')/I~ =0.53 and0.10 for p =3 and 4. To the extent that p +3 scat-tering may be neglected, the l =2 intensity will bequasiperiodic in y.We have numerically verified the validity ofEq. (4) by using Debye's localization principle"to modify Mie theory so that only partial wavesassociated with p =3 rays were included in theMie series. Furthermore, when Eq. (4) is ap-plied to spheres with certain m )1, the resultingIP'(y =0) agree with the glory "analog" tabulatedin Ref. 11. This analog was derived by applyingthe Watson transformation to the y =0 Mie series.Figure 2 diagrams the experiment. A syringeinjected bubbles into the liquid. The liquid had ahigh kinematic viscosity [= 600000 cS; 1 stoke (S)kE, exp[i(kR +q)]P 2~ g tel/2x f, sW' exp[ik(s —b)'/2n]ds, (2)I,' =(2/~)xI, f, , [W'(y, $)]',where I~ =I, a'/4R' is the total intensity at a dis-tance R =CQ from a perfectly reflecting sphereof radius a predicted by ray optics, ' I, is the in-cident intensity, and f~, =b'oi/a'q =b'(—n —a)/a'.In Eq. (4), R has replaced R' from (2) andy be-comes 180' —p because ~ »a. Geometrical op-tlcs3 9 g ves the lntensltles Ipl of separate axial27r~' =Jo F'exp[- iks siny cos(P —$)]dP, (3)where $ is the angle between e, and the projectionof C'Q on the exit plane. In Eq. (2), the approxi-mation given by Eq. (1) has been extended beyondits useful domain in anticipation of the stationaryphase approximation (SPA) of the integral. Directevaluation of Eq. (3) gives &'(y, () = &'(y, &,s =b)=~[(c, +c,)J'0(u) +(c,—c2)J,(u) cos2$] and ~2 =~(c,—c,)J',(u) sin2$, where u =kb siny. The SPA ofEq. (2) gives the Pth glory contribution to the scat-tered field when kb'/n, and thus x, are large. Inthe experiments to be described x - 4000 and theSPA is applicable.The total field may be approximated by summingthe E~' from Eq. (2) with the fields due to axialreflections and surface waves. Surface wave con-tributions shou1d be small for the observed bub-bles because of the largeness of x. To determinewhich glory and axial terms are important to thetotal field, and for other heuristic reasons, con-sider the l-polarized intensity I~' of the pth fieldtaken alone The SPA .of Eq. (2) givesBACKSCAT TEREDLIGHTPOLARIZERBEAM SPLITTERLASERBEAM(~} ' = I (b)BEAM CELLDUMPS BUBBLEL I QUIDiMIRRORFIG. 2. Apparatus for observing backscattering frombubbles.I degreeFIG. 3. Photographs for (a) crossed polarizer (l = 2scattering); (b) uncrossed polarizer (/ = 1); and (c) nopolar izer. The incident polarization was vertical.a = 0.49 mm and x= 6830.915VOLUME 47, NUMBER 1) PHYSICAL REVIEW LETTERS 28 SEPTEMBER 1/81a) IOCElIOK 00I It 2Bubble Radius ' (mm )FIG. 4. Measurements and model for the angularseparation of the dark rings in the E = 2 scattering.=1 cm'/sec] and a single bubble could be observedfor hours. The laser's power output was 5 mWand the beam diameter was 5mm. The wavelengthin the liquid, 2'/h, was (632.8 nm)/1. 403; e, layin the splitter's plane of incidence. The camerawas focused on so that the photographs record-ed the far-field intensity pattern. '" Photographswere made with a = 0.3-0.8 mm corresponding tox = 4000-11000. Exposure times were typically5 s for TriX film and a 200-mm-focal-lengthcamera lens.Figure 3 demonstrates that the scattering hasroughly the dependence on ( predicted by Eq. (4);( =0' corresponds to scattering toward the top ofthe photographs and y =0 corresponds to the cen-ter of the symmetry. Figure 3(b) shows that thel =1 scattering for y) 0.2 is significantly stronger for g =+ 90' than it is for g =0 . This agreeswith the following model results: (i) (c,/c, )'»1(for p =3 we predict c,/c, = —5.2); and (ii) for thisx, the I~' depend only weakly on $ and are domi-nated by the I~'. One prediction of Eq. (4) couldbe quantitatively checked: when both siny = y andu»1, the minima in I~' should be spaced by &yrad such that hb&y = ~, where for P =3, 5/a=0.447. Figure 4 compares this with the meanspacing of =40 dark rings lying outside the 9thring from the center. The error bars combineuncertainties in measured a and &y with those ofcorrections due to refraction at the cell-air inter-face' and the tilt of the cell. Figure 4 shows thati) =3 rays dominate the i =2 scattering. The modu-lations of the intensity along ( =+ 45' in Fig. 3(b)show that other rays contribute to l =1 scatteringsince the predicted I,' cc [&,(&)]'.In conclusion, backscattering from bubbles canbe enhanced by axial focusing. The number ofsignificant glory terms depends on m. The maincontributions differ from those for water dxopswhere surface waves' and other diffraction re-lated terms' play an essential role. If focusingwere not present, scattering by large bubbleswould be « In in the region' ' (p, +10') ~ y s 180',where p, =2 cos 'n = 89' for m ' =1.403. We alsofind evidence of j& =3 glory in Mie computationsfor bubbles in ~ates.This work was supported by the U. S. Office ofNaval Research. One of us (P.L.M.) is an AlfredP. Sloan Research Fellow.'G. Mie, Ann. Phys. (Leipzig) 25, 377 (1908).V. Khare and H. M. Nussenzveig, Phys. gev. Lett.33, 976 (1974).H. C. Van de Hulst, Light Scattering by Smal/ I'ar-tictes (Wiley, New York, 1957).H. C. Van de Hulst, J. Opt. Soc. Am. 37, 16 (1947).'V. Khare and H. M. Nussenzveig, Phys. H,ev. Lett.38, 1279 (1977).H. M. Nussenzveig and W. J. Wiscombe, Opt. Lett.5, 455 (1980).~P. L. Marston, J. Opt. Soc. Am. 69, 1205 (1979).D. L. Kingsbury and P. L. Marston, J. Opt. Soe.Am. 71, 358 (1981).G. E. Davis, J. Opt. Soc. Am. 45, 572 (1955).M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).~~J. J. Stephens, P. S. H, ay, and T. W. Kitterman,Appl. Opt. 14, 2169 (1975).J. W. Goodman, Introduction to I"ou,~ex Optics(McGraw Hill, New York, 1968).916

Glory in optical backscattering from air bubbles

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Glory in optical backscattering from air bubbles

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