382 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER Thermal Convection and Spherical Aberration Distortion of Laser Beams in Low-Loss Liquids Abstract-Experiments and approximateanalysis are given for spherical aberration interference rings and for convection distortion arising in the thermal self-defocusing effectsproduced by laser beams passing through materials of finite loss. The importance of thesethermal effectsformoderate-power beams in materials of relatively low loss is stressed. Thermal gradients arising from heat conduction have been shown to produce a self-defocusing effect on laser beams passing through materials with finite loss.P].[21 The effect is observable even when loss is small, and has consequently been usedI31-I~lfor the measurement of absorption coefficients as low as 10-4 cm-’. Kiefer and Brault[el also stressed the difficulties this defocusing may cause when a laser beam of moderate power passes through a materiat of small but finite loss. They observed undesired spreading or “blooming” of the beam from a CW argon ion laser when the beam passed throughan indexmatching fluid. This correspondence describes the interference patterns arising from the fourth power term (the first term contributing to spherical aberration) in the power series expansion of optical length for suchlenses. It also describes thermal convection distortion of the beam when the absorbing material is a liquid. Observations of these effects are made by passing the beam from a CW ionlaserthrough a cell containingnitrobenzene. It becomes obvious that thethermal effectsdescribed can cause serious distortions of laser beams of moderate or high power when passed through many materials that are normally thought to be transparent. We describefirst the experimentalresult. Thearrangement is CW argonionlaseris sketched in Fig. 1. The beamfromthe obstructed by a card placed betweenthe laser and thecell containing nitrobenzene for a period long enoughto dissipate all thermaleffects. The cardisremovedandthespot is firstseen as inFig. 2(a), corresponding tothe normaldiffractionspreading of the beam. At t = 1 s [Fig. 2(b)], the spot is expanded to about double its original size, and shows definiteinterference rings. At 1 = 2 s [Fig. 2(c)], the beam has reached nearly its maximum size, about three times theoriginal diameter, and here also the rings are clearly present. At t = 3 s [Fig. 2(d)], a flattening of the top is observable, and this flattening continues to develop in the pictures for t = 4 s [Fig.2(e)] and t = 5 s [Fig. 2(f)]. Thepicture issubstantially unchanged for later times. The timedevelopment of Fig.2 thus shows three effects. The firstis the thermal conduction self-defocusing referred to in the firstparagraph, The secondis an interference effect which we believe to be caused by the spherical aberration of the thermal lens. The third is the flattening which appears to be caused by thermal convection gradients arising from the absorption heating from the laser beam. I n order toestimatetheamount of sphericalaberration,we utilize the analysis of Gordon et al., P I which gives an expression for temperatureincrease as a function of radiusandtime when a Gaussian beam is introduced into a medium a t time zero, as follows: CW Argon laser 1- P = k T = = t = wn = absorption coefficient of material, cm-l total power in laser beam, watts thermalconductivity of material, cal/cm~s”K radiusfrom axis, cm time after turning on beam, s radius a t which Gaussian field decreases t.o e-l the axis value Manuscript received July 18,1967. This research was supported by the National Science Foundation under Grant GK-457. 180cm 41 cm I---TEM,, 235 cm J mode Sample cell length 5 cm Fig. 1. Experimental arrangement,. Fig. 2. D p Time development of beam pattern a t screen position after injection of laser beam into nitrobenzene cell. = k/pCp = density,g/cm3 cp = specific heat, cal/g. OK Ei = exponential integral. The steady-state result for a beam in a cell with the outer radius a held at constant temperature is also given as: As in Gordon et al., P I we plot (1) and (2) in Fig. 3. It is seen that the temperature variations with radius approach the steady-state value ina time of the order of T~ = w02/4D, which is about 2 seconds for the organic liquids studied in Gordon et aZ.[ll Fig. 3 also shows that a square law approximation is valid up to about T = WO.Since 87 percent of the energy is contained within that radius, the main effect is that of a defocusing lens. There is, nevertheless, a measurable amount of light beyond Wg, so it is desirable to expand (2) at least to terms in 1.4.The result is: AT(?.) M !LE IC E [In (?E$ - 2(;)5 + (31. (3) The spherical aberration arising from terms in 1.4 produces interferenceringsin the far-zonepattern.171 Using the nomenclature of Born and Wolf, [71 @ = = Sample Power out = 320 mW where b 1967 [optical path length to terms in r4] - [optical path to terms in rz] = const X A(r/Tn) (4) where y o is the aperture radius. The aperture for our problem is somewhatarbitrary since the Gaussianbeam does nothavean abrupt edge, but it should be somewhat greater than the measure wo,say 1.2 WO. The calculated value of the spot size at the position of the cell is 0.7 mm, in good agreement with the observed beam diameter. 383 CORRESPONDENCE Fig: 4. Effect of fluid convection showing blurring of temperaturegradient near the’top of the beam. wo values give the maximum upward velocity from thermal convection as 2.4 mm/s. The upward motion of the liquid is dominant at the top of the beam, and smears outthe index gradient there, as indicated in Fig. 4. Our estimates of the velocities of s,cattering particles in ‘Theabsorption coefficient of the liquid,calculatedfrom the de- the cell were around 1 mm/s, which would also be consistent with focusing effect of the beam and eq. (10) of Gordon et al., yields the two or three second time required to stabilize the convection .b =12.4.10-3. In this calculation and others to follow, wavelength flattening of the beam after thermal gradients are established. The ismtakenas the dominant 4880-A line of the argon laser, although approximate calculat.ion made here is thus of the proper order of ather frequencies were present. It should also be noted that the magnitude. thin lens approximation is marginal because of the high power and Callen et al.[81 observed interference rings and a distortion of the the relatively long cell used, but is useful for the order-of-magnitude beam when the output of a few milliwatt helium-neon laser was results presented. Interpretation of (4) by the use of (3) gives passed through a 10-cm length cell containing carbon disulfide with avanadiumpthalocyaninedyeadded. The beamdistortion wits attributed to misalignment in that experiment, and it does seem unlikely that convection effects would play a part at those power caused by the effect For an/aT = 10-3, 1 = 5, P = 0.32, k = 3.9 X and X = levels. The interference rings are very likely describedin this correspondence,although the greaterlength of 0.488 x 10-4, their cell complicates the analysis of the effect. Prof. 111aba[~l has observedboththeconvectioneffectsandinterferenceringsin @ M 25 x 1.2w0 ( q 4 liquids placed inthe output of C o n lasers. Materials with extremely low losses are likely to show these thermal distortions in the highFig. 9.4 of Bornand gives a plotfor rI, = 16 X ( ~ / r o ) ~which , power COL.beams,andmaterialswithmoderate losses, normally showsabout 20 rings. Fig. 2(f) shows about 13 rings. Thus, con- thought of as being transparent, will show the effects in the few watt sidering the uncertainties of apertureandabsorption coefficient, level beams commonly produced by CW ion lasers. the agreement is reasonable andthe effect seems to be the spherical ACKNOWLEDGMENT aberration postulated. I n order to estimate the effect of thermal convection inthe liquid, Theauthorsgratefully acknowledge valuablesuggestionsfrom the temperature distribution may be taken from (2) and a change in E. Ippen of The ElectronicsResearchLaboratory,University of density arising from thermal expansion calculated. This provides a California, Berkeley, and special help with the experimental appagravity force, which with the viscosity equation, allows calculation ratus from E. Clausen. of a velocity distribution. The problem has not been solved exactly, J. R. P[’HINNERU but estimate of an upper bound has been made by approximating D. T. MILLER (2) byaGaussianformanddividing the beamintodifferential F. DABBY vertical strips. The gravity force on each strip, by integration in University of California the vertical direction, is Berkeley, Calif. rBig. 3. Theoretical temperature-gradient as a function of radiusandtime, from Gordon et ~ 1 . 1 1 1 REFEREXCES where p = density, CY = expansion coefficient, AT, = maximum temperature rise, and g = acceleration due to gravity. The velocity equation in an incompressible fluid with viscosity ,u is then Integration of this equation yields a resultfor velocity on the axis V,(O) = f f g AT,TW2, ----. 1 6 ~ The viscosity of nitrobenzene is about 2 centipoise. The maximum temperature differential for the values used earlier is 0.5”C. These J. P. Gordon R. C.C. Leite, R. S. Moore, S. P 8 Porto, and J. R. Whinnery, “Long transient ekects in lasers with inserted 1iq;id samples,” J . A p p l . Phzls., vol. 36,pp. 3-8, January 1965. [21 K. E. Rieckhoff, “Self-induced divergence of CW laser beams in liquids-a new nonlinear effect in the propagation of light,” A p p l . Phys. Lett.. vol. 9,p. 87, 1966. 131 R. C. C. Leite. R. S. Moore, and J. R. Whinnery, “Low absorption measurements by means of the thermal lens effect using an He-Ne laser,” A p p l . Phys. Lett., vol. 5 , 141-14:; October 1 1964. [dl D. Solimini, .4pcuracy hnd sensitivity of thethermal lens method for measuring absorption, Appl. Optics, vol. 5, pp. 1931-1939, December 1966. [ K l D. Solimini, “Loss measurements of organic materials a t 63288,” J . A p p l . Phvs., vol. 37,pp. 3314-3315,July 1966. [SI J. E. Kiefer and R. G. Brault. “High power effects on optical fluids,” presented a t 1966 Conference on Electron Device Research, Pasadena, Calif. (referenced by permission). 171 M. Born and E. Wolf, Principles of Optics. New York: Macmillan, 1964, sec. 9.4.1. [SI W. R. Cullen. E. G. Ruth, and R. H. Pantell “Thermal self-defocusing of light ” Stanford University Mmowave Laboratory. Rept. 1489,December 1966. 19) H. Inaba, Tohoku University, private communication. b[ll