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Two photon absorption, nonlinear refraction, and and optical optical limiting in semiconductors Eric W. W. Van Van Stryland H. Vanherzeele A.Woodall M. A. Woodall Soileau M. J. Soileau Arthur L. L. Smirl Shekhar Guha F. Boggess Boggess Thomas F. North Texas Texas State State University Center Electronics Center for Applied Quantum Electronics Physics Department of Physics Denton, Texas Texas 76203 76203 Abstract.Two Two-photon Abstract. -photon absorption coefficients /32 ß2 of ten direct gap gap semiconducsemiconductors with band-gap with band -gap energy Eg Egvarying varying between between 1.4 and 3.7 eV were measured pirn and and 0.53 pm ^m picosecond picosecond pulses. j32 found to to scale scale as as E93, E~3 , as using 1.06 pm ß2 was found predicted by the samples samples measured. measured. Extension Extension of empirical predicted by theory theory for the of the empirical f32 and Eg Eg to Eg = relationship between ß2 agreeto InSb InSb with with Eg = 0.2 eV also provides agreepreviously measured measured values values and the predicted ß2. /32 . In addition, the ment between previously absolute values of ß2 p2 sre are in in excellent agreement (the average average difference difference being being <26%) includes the the effects effects of of nonparabolic nonparabolic bands. bands. <26 %) with with recent theory, which includes induced in these materials was monitored and The nonlinear refraction induced and found found to agree assumption that the selfself-refraction agree well well with the assumption refraction originates originates from from the the two-photon-generated observed selfself-defocusing two -photon -generatedfree free carriers. carriers. The The observed defocusing yields yields an effective nonlinear nonlinear index index as as much much as as two two orders orders of of magnitude magnitude larger larger than than CS2 CS2 for comparable comparable irradiances. irradiances. This self-defocusing, two-This selfdefocusing,inin conjunction conjunction with two photon absorption, was used used to to construct construct aasimple, simple,effective effectiveoptical opticallimiter limiter that that has high input irradiance irradiance and and low low transmission transmission at at high high has high transmission transmission at at low input irradiance. The The device is the optical optical analog analog of of aa Zener diode. input irradiance. Subject terms: terms: nonlinear nonlinearoptics; optics;two two-photon effects; index index of of refraction; refraction;semiconductors. semiconductors. Subject -photon effects; Optical Engineering 24(4), 1985). 24(4), 613-623 613 -623 (July/August (July /August 1985). CONTENTS 1. Introduction Introduction 1. 2. Theory Theory 2. 3. Experiment Experiment and data 3. 3.1. Experiment 3.1. Experiment 3.2. Data Data 3.2. 4. Comparison of of ß2 /32 values values to theory 4. Comparison Two-photon theory 4.1. Two -photon absorption theory 4.2. Comparison 4.2. Comparison to to theory theory 5. Self-refraction 5. Self -refraction Optical limiter limiter 6. Optical 7. Conclusion 7. Conclusion Acknowledgments 8. Acknowledgments References 9. References 1. INTRODUCTION 1. INTRODUCTION specific material give rise to these high high nonlinearinonlinearispecific material parameters that give ties. This This predictive predictive capability capability is is extremely extremely important important from from the the ties. of searching searching for materials with large nonlinearities. standpoint of A study of the the nonlinear nonlinear optical properties of several several semiconducsemiconducpresented here, and aa relationship relationship between between the the two two-photon tors is presented -photon absorption (2PA) coefficient coefficient £t2 material properties properties isis absorption (2PA) ß2 and and other material verified. Ten materials were were experimentally experimentally studied verified. Ten different different materials studied for photon energy1 energy^ band-gap which the incident photon w is less less than than the direct band -gap energy Eg but greater greater than than EEg/2 thattwo two-photon absorption isis energy E but /2 sosothat -photon absorption allowed J rBoth Both 1.06 1.06 and 0.53°µm O.SJ/xm picosecond picosecond pulses pulses were were used used in in allowed. transmission experiments ranging transmission experiments using using semiconductors semiconductorswith with Eg E ranging 1.4 to 3.7 3.7 eV. eV. We We find find that thatthe the2PA 2PAcoefficient coefficientß2/^isisgiven given by from 1.4 = Ks/Ep K v E f(2ftg»/Eg) f(2$0/ Eg) (1) n2E3 g ever-increasing in light light-wave The ever -increasing role of semiconductors semiconductors in -wave technology characterization of of the the nonnonhas created a pressing demand for the characterization linear optical properties properties of of these these materials. materials. Semiconductors Semiconductors are are linear optical as elements elements in nonlinear nonlinear optical optical devices devices because because of their attractive as large nonlinearities. A A careful careful large and and potentially extremely fast optical nonlinearities. study of these these macroscopic macroscopic nonlinearities should allow one to determine micro mine the the dependence dependence of of these these nonlinearities nonlinearities on on fundamental fundamental microscopic scopic mechanical and and electronic electronic material material properties (e.g., band gap, carrier lifetime, carrier base formed formed by by carrier effective mass, etc.). The data data base this information would only to tabulate tabulate the the this information would then then allow allow one one not not only materials exhibit large large nonlinearities nonlinearities but predict the the materials that that exhibit but also also to predict where K is is aa material-independent material- independent constant, n is the linear refractive is nearly materialmaterial-independent index, and EEp is independent for a wide variety of function f,f, whose whose exact form form depends depends on the semiconductors.* The function structure, is is a function only of the assumed band structure, the ratio ratio of the the photon photon energy ftcu are optically optically energy fiw to to Eg Eg,, which which determines determines the the states states that are coupled. The scaling scaling given (1) agrees coupled. given by by Eq. Eq. (1) agrees with with the the most recent two-photon absorption3-5 3 " 5 and allows for predictions of of theories for two -photon absorption coefficients for materials at at other otherwavelengths wavelengths given given 2PA coefficients for other materials parameters. For Forexample, example,extension extensionof ofthis this scaling scaling minimal material parameters. 10.6 Mm fa of 6.8 W, [see Eq. (21)] (21)] to to InSb (300 K) at 10.6 µm predicts a ß2 6.8 cm/ cm/ M MW, Specifically, which is in in excellent excellent agreement agreement with recent experiments. Specifically, Miller et al. W. Equation (1) is therefore al.66 obtained obtained a value of 88 cm/ cm/ M MW. Equation (1) therefore Invited Paper Paper NO-110 NO-110 received received Jan. Jan. 15, 15, 1985; 1985; revised revised manuscript manuscriptreceived receivedFeb. Feb.7,7,1985; 1985; accepted 7, 1985; 1985; received 1985. accepted for publication Feb. 7, received by by Managing Managing Editor April 8, 1985. © 1985 1985 Society Engineers. © Society of of Photo-Optical Photo -Optical Instrumentation Instrumentation Engineers. *E ==2P2m 2P2 m/fi2 where PP isis the the Kane Kane momentum momentum parameter parameter and m m is is the the *E /ft2,, where mass (Ref. 2). 2). electron mass OPTICAL ENGINEERING / July/August 1985 / Vol. 2424 No. 4 /4 / 613 OPTICAL ENGINEERING / July /August 1985 / Vol. No. 613 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx VAN VAN STRYLAND, STRYLAND,VANHERZEELE, VANHERZEELE,WOODALL, WOODALL,SOILEAU, SOILEAU,SMIRL, SMIRL,GUHA, GUHA,BOGGESS BOGGESS valid over a range of 20 in band-gap band -gap energy energyfrom from the the infrared infrared to the visible. visible. In In addition, addition, we we find find that that the the proportionality constant K K as calculated by Weiler Weiler44 for fornonparabolic nonparabolic bands agrees agrees with with our our experexperimentally determined K K to to within within better betterthan than26 26%. %. We We used used the experimentally determined determined 2PA coefficients along coefficients along with Drude theory theory (modified (modified to toinclude includeeffects with a modified Drude effects of ofinter inter-band transitions and band band filling) filling) to model the nonlinear refraction in in these these semiconductors. semiconductors. We quantitatively fit the predictions of this theory to beam beam propagation data obtained obtained for for CdSe theory CdSe and and obtain obtain refraction isis assumed excellent agreement when all of the nonlinear refraction arise from from the the carrier carrier generation.' to arise generation. 7 That That is, is, the thecontributions contributions proportional the photogenerated photogenerated carrier carrier density proportional to the density dominate dominate the the electron nonlinear nonlinear refractive refractive index usual bound electron index changes. changes. This has previously shown been previously shown to be the case for one -photon absorption absorption in one-photon /im 8 and and Si Si at at 11 µm.9 pm. 9 We find materials such as InSb at 55 µm8 find that the effective nonlinear effective nonlinear refraction can be two two orders of magnitude larger that for for CS2 CS2 at comparable comparable irradiances. irradiances. than that combined effects effects of of two two-photon Finally, we utilized the cohibined -photon absorption and nonlinear nonlinear refraction refraction in in GaAs GaAs totomake makean anirradiance irradiance(flu (flu-ence) device. 10 This ence) limiting device.03 Thisdevice devicehas has high high linear linear transmission at low irradiance irradiance (fluence) (fluence) and low and low transmission transmission at high high irradiance irradiance (fluence). high irradiances, irradiances, laser (fluence). At At very high -induced melting melting is is also laser-induced This device device is passive, involved in the limiting action. This passive, has picosecond turn-on turn -on time, time, and and isis the the optical optical equivalent equivalent of a Zener diode. Sec. 2 we we describe the model model used used and and derive In Sec. derive the the equations equations needed needed to to describe describe both both the the nonlinear nonlinear transmission transmission and the nonlinear refraction semiconductors studied. refraction observed observed in in the the semiconductors studied. In In Sec. Sec. 3 we we outline experimental procedure procedure used used to determine determine the outline the the experimental the two two-photon absorption absorption coefficients coefficients and present present the the experimentally experimentally photon determined 2PA 2PA coefficients. Section 4 presents a comparison comparison of the 2PA data to theory for parabolic and nonparabolic nonparabolic bands bands with with and without exciton corrections. In In Sec. Sec. 55 we we present present the the experiments experiments propagationdata datausing usingthe theresults resultsof ofSec. Sec. 2. and fits to the beam propagation 2. We We describe aa semiconductor semiconductor optical describe optical limiter limiterbased basedon on 2PA 2PA and and its its design, operation, and uses uses in in Sec. Sec. 6. 6. design, THEORY 2. THEORY The experimental configuration used used throughout throughout this work was was one one in which which the sample sample was was very very thin thin compared compared to the confocal beam and moreover, moreover, any anyself self-induced beam phase parameter, and -induced beam phase changes changes were small effects in the sample were small enough enough that that beam propagation propagation effects were self-action negligible (i.e., selfaction was was"external,"as "external," as described in Ref. 11). 11). In this case case the Maxwell wave this the Maxwell wave equation equation for for the the propagation propagation of the electric electric field field EE can be written as 2ik ôE ôz = w2 iwaµo E- -c-12 X(3) I E I2 E, (2) where X(3) x^ denotes denotesthe the third-order third -order nonlinear nonlinear susceptibility and iNe2 n 1\ /^T — 4- iNe2 a =-(a(a++aeXN) aex N) n -+ V µo ^o mehw meh w (3) Here, we we have have explicitly included denotes the conductivity. Here, included the possibility throughthe the term term aeXN, sibility of ofphotogenerated photogenerated carrier absorption absorption through aexN, carrier cross cross section section (holes (holes plus electrons) and where aex is is the total carrier and is the density N is density of these these carriers. carriers. Also, Also, a is usual residual residual linear linear is the usual (e.g., band band-tail absorption (e.g., -tail absorption, absorption, impurity impurity absorption, etc.), etc.), meh isis the the reduced reducedelectron electron-hole effective mass. Writing and meh -hole effective Writing the the electric field as E = = Ae'4' Aei<J> E (4) with the irradiance given by I == (ne0C (ne0 C/2)A2 with /2)A2,, Eq. Eq. (2) (2) can can be sepasepagiving rated, giving dI -aI --ß2I2 £ = -ol /3212 --aeXNI aexNl dz = (5) and dcl) dz = ßiI - (6) /32 , thethe two-photon absorption where 1632, two -photon absorptioncoefficient, coefficient,isisproportional proportional to to part of of X(3) x^ and { == wy/c the imaginary part and Pß1 wy /c isisproportional proportional to the real part of ofX(3). x (3^- 7y is the more moreusual usualn2 part is related related to the n2 by by n2 n2 (esu) == en en y/ y/40 40 rr, TT, where the right-hand right -hand side side of of the the equation equation is is in in mks mks system systemunits. units. yy!l in in (6) is is given by Eq. (6) by Mo1e2 CP y\y, _= µoe2CP 2nmehw 2nmej1 a> (7) (7) The parameter P is is introduced introduced here here to to account account for for contributions contributions to the nonlinear refraction proportional proportionalto toNex Nex but not not explained by by the Drude model. An example of such such aa contribution contribution is that arising from interband transitions. transitions.12 12 interband The equation governing the carrier generation is is dN dt ft* /3212 (8) 2Yiw showing that for every two absorbed photons -hole pair photons one oneelectron electron-hole is is generated. generated. This equation is valid only for pulses pulses short short enough that recombination and diffusion can be be neglected neglected during the the pulse. pulse. We We assume this to be be the the case case for for our ourpicosecond picosecondpulses.13 pulses. 13 We note note that Eq. (5) (5) for for the the irradiance irradiance is is independent independent of the phase (6)]. This assumption of of aa thin [Eq. (6)]. This is is due due to our assumption thin sample sample (i.e., (i.e., no no irradiance changes changes due due to to nonlinear refraction within within the material). We We can, can, therefore, solve Eq. (5) simultaneously with Eq. Eq. (8) (8) for the beam attenuation in aa sample sample of of thickness L. These equations must attenuation in be be solved solved numerically numerically unless unlessthe thecontribution contribution to to the the absorption absorption from the photogenerated carriers is is negligible. negligible. We the photogenerated We can estimate under under what conditions this is true by finding finding the the irradiance, irradiance, denoted by by 'Cr' Icr , for which the the carrier absorption is equal equal to the the multiphoton absorption. approximate relation is is found limit of of small tion. An approximate found in the limit small total as absorption as /2-Rqj Jcr ~" 2 V2 tub 'Cr aeXto(1 - R) (9) where R R is is the surface surface reflectivity reflectivityand andtot0isisthe theHW H W11 // ee M (half (half-width -width at 1/e 1 / eofofthe the maximum maximum inin irradiance) irradiance) of of the assumed assumed Gaussian Gaussian temporal profile profile pulses. pulses. This result was first given given by Bechtel Bechtel and Smith. 13Note Notethat thatthis thiscritical criticalirradiance irradianceisis independent Smith." independent of of02 /32 since since both the transmission change and the photogenerated carriers result materials with with small small ß2, /32 , which require from 2PA. Thus, materials require high high incident incident irradiance to observe a transmission change, will will be be the most likely likely affected by materials to be affected by photogenerated photogenerated carrier absorption. The contribution to the change in transmission from these these carricarriers is is proportional proportional to tot. t~ l . Longer Longer pulses pulses of of the the same ers same irradiance irradiance more energy and, therefore, produce contain more produce more more carriers. carriers. We We can can absorpdetermine if these these carriers carriers are are contributing to the nonlinear absorpby measuring measuring the change change in transmission transmission for tion by for different different pulse pulse We show in Sec. used are widths. We Sec. 33 that that the irradiances irradiances used are well well below Icr that we can ignore and that ignore photogenerated photogenerated carrier absorption. absorption. This This is is the reason for using picosecond pulses, pulses, as as discussed discussed in in Ref. Ref. 13. 13. While While we find the carrier carrier absorption absorption to to be be negligible, negligible, the the refractive refractive index find index proportional to to the the carrier carrierdensity density[Eq. [Eq. (6)] (6)] isis definitely definitely not change proportional not negligible, as discussed in Sec. negligible, Sec. 5. 5. The solution to Eq. Eq. (5) (5) with this assumption assumption isis I(z,r,t) I(z r t) = R)J I(o,r,t) I(o,r,t)e~az (1 -- R)i e az 1 + +q(z,r,t) q(z, r, t) (10) ( ) where - -R)R) (1 (1 -e~az)/a. whereq(z,r,t) q(z,r,t)==ftl(o,r,t) ß21(o,r,t)(1 (1 -e)/a. Inside Inside the the 1, and behind the sample j = 22 since there are two surface sample jj = 1, 614 / OPTICAL / OPTICALENGINEERING ENGINEERING / /July/August Vol. 24 July /August 1985 1985 // Vol. 24 No. No. 44 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx SEMICONDUCTORS IN SEMICONDUCTORS REFRACTION, AND OPTICAL LIMITING IN TWO PHOTON ABSORPTION, NONLINEAR REFRACTION, effect The effect sample. The the sample. of the is equal to L, the length of reflections. Also, z is but ignored but been ignored has been absorption has of the rear surface reflection on the absorption /32 determinationofofß2 the determination in the errors in is not expected to lead to significant errors give the used. 14 Equation for the samples used.14 Equation (10) (10) can can be be rearranged rearranged to give of a sample of position of each radial position instantaneous transmission T' at each length L as [l+q(L,r,t)]e«L [1 + g(L,r,t)] eaL T'" 1 of T' Since q(z, r, t) isis directly directly proportional proportional to to I(o,r,t), I(o, r, t), aa plot of Since q(z,r,t) 1 intercept whose intercept line whose straight line yield a straight should yield versus incident irradiance irradiance should pulses /32 . Experimentally, pulses whose slope determines ß2. and whose determines aa and requires which requires used, which are used, profiles are temporal profiles of Gaussian spatial and temporal Eq. (10). Taking into account the of Eq. integrationof temporal integration spatial and temporal temporal and spatial integrals, find for the pulse transmission integrals, we find •s< MONITOR MONITOR .,l ^ g_-A - -5-^ v-H s't-i-HAMP F [ AUD I \ » J A 1 ATTEN. ATTEN. ! tp MONITOR MONITOR (ID (1 --R)2 (l R)2 ENERGY ENERGY Nd:YAG Nd:YAG SYSTEM LASER LASER SYSTEM n >" i 1—1 COLLIMATOR COLLIMATOR SAMPLE FF SAMPLE -Q- ------ -.{|- -- -Q—>-| --- -CDxy ••'' // ^VIEWER ,/VIEWER ENERGY ENERGY MONITOR MONITOR HeNe HeNe absorption two-photon the twomeasuring the setup for measuring Fig. 1. Experimental photon absorption Experimental setup Fig. 1. indicates splitter indicates beam splitter the beam before the arrow before The arrow /um. The 1.06 µm. /J2 at 1.06 coefficientsß2 coefficients were pulses were /xmpulses 0.53 µm second-harmonic the secondofthe position of the position harmonic crystal when 0.53 used. by given by fluence, given oo 00 2a(l 2a(1 -R) - R) = T T= %/TT ß2(eaL _ 1) -xz] q(L,o,o)e~x2] e $dxln[l +q(L,o,o) (12) 0 '<••'••>- ^ - Io exp to (13) )2 (1 --R) tit [1 + q(L,r,t)] <J>(o,r,t)+ 3>(L,r,t) = (1)(o,r,t) (I)(L,r,t) + ß2 (1 02 (1 - R)2 71 t 5dt' Fl (t') 2fi (14) , -00 where - aBn[1 +q(L,r,t)] F,(t) = Fl(t) +q(L,r,t)] é al 1 e~aL q(L,r,t)a I" _ _ q(L,r,t)q + q(I-, r, t) ' 11 +q(L,r,t)J 11 -e~«L -é aL [1L (15) the describe the completely describe together completely (14) together and (14) 2) and Equations (10) (10) (j(j = 2) Equations electric field field at at the the exit exit plane plane of of the the sample. sample. From From these solutions for sample the sample outsidethe positionoutside anyposition fieldatatany 4>(L,r,t), andc(L,r,t), I(L,r,t) and thethe field I(L,r,t) Huygens-Fresnel usingthetheHuygens determinedusing then canthen z,r,t)can (L +-f z,r,t) bebe determined -Fresnel 15 formalismasas15 propagation formalism propagation 27r E L+ z,r,t)=-exp z,r,t = ex J' r irrr2 ;rr \ Xz J ¡ r'dr'EE (L,r',t L,r',t-- -) exp (-XzX X J r'dr' 00 c?. J z,r,t)| 2 dt . (17) . z t7rr'2 Experimental results results are are compared compared with with numerical numerical evaluations evaluations of of Experimental (17), as described in Sec. 5. and / or spatial spatial integrals integrals of Eq. Eq. (17), (17) and/or Eq. (17) EXPERIMENT AND DATA 3. 3. EXPERIMENT T"-11 versus The resulting plot of T versus II has has a slight slight downward curvature the both the caused by these integrations, since at the higher irradiances both the of the rear of spatial and temporal profiles are broadened toward the rear sample (i.e., there there is is more more 2PA 2PA at at the middle, middle, brightest brightest part of the sample (i.e., beam). Examples function of I0 Io are of TT"-11 from Eq. (12) as aa function plots of of plots Examples of 3. Sec. 3. of Sec. 5(b) of shown in Figs. 5(a) and 5(b) propagabeam profile of the pulse and its propagaIn order to model model the beam 1) 1 0) (j(j ==1) (6) using Eq. (8) for N and Eq. ((10) Eq. (6) tion, we integrate Eq. we now integrate phase: thephase: for the expression for an expression obtain an for the irradiance to obtain + F(L F(L ++ z,r) z,r) = -00 where we have taken 1(o,r,t) = co Jo J0 ( 27rrr' Xz ) (16) o pulses, isis the experiments, using measure in we measure What What we in our experiments, using short short pulses, the 3.1. Experiment of transmission of the transmission measured the we measured experiments we ofexperiments set of first set the first In the several semiconductors semiconductors as as aa function function of incident incident irradiance irradiance to deterseveral experimental The experimental coefficients. The absorption coefficients. nonlinear absorption mine their nonlinear mine microwas a microused was source used laser source 1. The laser Fig. 1. in Fig. shown in arrangement isis shown arrangement processor- controlled, passively passively mode-locked mode -lockedNd:YAG Nd:YAG laser laser that processor-controlled, amplified pulses pulses of of energy energyup up to to 77 mJ mJ per pulse pulse at at single amplified produced single could width could pulse width The pulse mode.* The TEM 00 mode.* 1.06µm operated in in the TEMoo jum when operated 1.06 selecting etalons 150 ps be varied 40 and and 150 ps (FWHM) by selecting etalons of between 40 varied between was pulse was each pulse ofeach widthof Thewidth coupler. The output coupler. varying thickness as the output the ofthe energyof theenergy squareofofthe monitored thesquare ofthe ratioRRof the ratio measuringthe by measuring monitored by µm) pulse pulse to to the energy energy of of the the second second harmonic harmonic (1.06 pim) fundamental (1.06 crystal. 16 This LiIO3 crystal.16 in aa LilO3 produced in was produced (0.53 jum) µm) pulse that was This ratio is (0.53 the width, provided pulse width, laser pulse the laser directly proportional proportional to to the provided that that the directly by calibrated by was calibrated unchanged. This remains unchanged. spatial This ratio was profile remains spatial profile second-background-free nearlybackground usingnearly width using pulse width measuring the pulse -free second having aa pulses having onlypulses acceptingonly whileaccepting scanswhile harmonic autocorrelationscans harmonic autocorrelation the preset preset value. value. To To ensure ensure that that the ratio 15% of the fixed ratio R within 15% were scans were autocorrelation scans width, autocorrelation pulse width, R was proportional to the pulse to the ratio indeed, the ratio performed for three three output coupler etalons, and indeed, performed for scan isis autocorrelation scan such an autocorrelation example of such scaled properly. An An example scaled properly. shown in Fig. 22 along along with with the the best best Gaussian Gaussian fit. fit. The The autocorrelation autocorrelation pulse width of 38 Gaussian pulse to a Gaussian width of 54 54 ps ps(FWHM) (FWHM) corresponds to 38 ps (FWHM). ps(FWHM). temperature-tuned required, aa temperature was required, When 0.53 jum µm light was -tuned CDA When 0.53 (cesium dihydrogen arsenate) arsenate) crystal crystal was was placed placed in in the the beam beam at the (cesium dihydrogen was p,m was 1.06 µm at 1.06 Light at 1. Light Fig. 1. position by the the arrow in Fig. indicated by position indicated 100% dielectric reflecting mirrors. two 100% and two polarizer and blocked with a polarizer second-harmonic the second Autocorrelation -harmonic beam performed with ofthe scans of Autocorrelationscans crystal dihydrogen phosphate) crystal an angle -tuned KDP (potassium (potassium dihydrogen angle-tuned by divided by width divided pm pulse width 1.06 µm the 1.06 as the scaled as pulses scaled these pulses showed that that these Gaussian-shaped for Gaussian expected for v2 %, as -shaped pulses. Again as expected within1010%, \/2totowithin were as clean data were held fixed, fixed, and and the the autocorrelation autocorrelation data the ratio R was held 2. Fig. 2. 1.06 jum as those shown for 1.06 µm in Fig. the used in the were used 150 ps were Two different pulse widths widths of 40 40 and 150 different pulse the Since the /zm. Since 1.06 µm. transmission transmission experiments experiments on on each each sample sample at at 1.06 change the pulse width were to change used to etalons used coupler etalons output were optically optically output coupler well as the as well lineas beamline thebeam plate,the quartzplate, rotatable quartz contacted to a flat rotatable in change in percent change few percent (Afew fixed.(A remainedfixed. measured beam parameters parametersremained the in the self-focusing slightselfbyslight causedby probablycaused width probably focusing in spatial width the beam spatial Calif. **Quantel Quantelmodel modelYG40, YG40,Quantel QuantelInternational, International, Inc., Inc., Santa Clara, Calif. 4 /4 / 615 No. 2424 / Vol. 1985 / July/August OPTICALENGINEERING ENGINEERING / July /August 1985 / Vol. No. 615 OPTICAL Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx STRYLAND, VANHERZEELE, GUHA, BOGGESS BOGGESS VAN STRYLAND, VANHERZEELE, WOODALL, WOODALL, SOILEAU, SMIRL, GUHA, 1.2 1.0 1.0 0.8 si L 0.5 \ 0.0 00.. -80 -40 0 40 I °'4 UJ 0.2 0.0 80 80' -0.2 _ •A******^ -3 -1 DELAY (Ps) |ps) DELAY Fig. Autocorrelationscan scanofofpulses pulseshaving havingaa FWHM FWHM of of 38 38 ps ps as as calcuFig. 2. 2. Autocorrelation lated Gaussian (solid line) lated from the best fit fit Gaussian line) autocorrelation. amplifier was relative error error bars bars between between was taken into account.) The relative one transmission experiment the next, where one transmission experiment and and the where only the pulse pulse width was changed, were very small. While at high irradiances (a few irradiances (a G W/ ) we W /cm2 cm2) wedid didsee seea asmall smallpulse pulsewidth widthdependence dependenceof ofthe thetransmistransmission in some samples, this difference was consistent with values for free-carrier sections (10 (10~-17 17 to 10 10~-18 18 cm2 ). No the free -carrier cross sections cm2). No pulse pulse width at the the low low irradiance irradiancelevels levels (0.5 (0,5 GW GW/cm2 dependence was observed at /cm2 )um) used used to to extract extractvalues values of of the the 2PA 2PA coefficient. at l1 µm) coefficient. A A calculation of Icr Sec. 22 for fortypical typicalsamples samplesatat1µm 1 ^im Icr from Sec. gives IcrIcr ---~5 5 GGW/cm2 W / cm2 gives = 55 X 10 10~ for aex = 18 cm -18 cm2.. In In fact, fact, Eq. Eq. (9) (9)considerably considerably underestimates underestimates Icr Icr isis several several times times 'Cr' From computer calculations, we find that Ier larger, the difference difference arising larger, arising mainly mainly from from the the fact fact that the the spatial spatial averaging was ignored in the approximate expression. irradiance averaging expression. In In addition, at 0.5 0.5 µm jum the the contribution contribution of ofphotogenerated photogenerated carrier carrier addition, will be be less less than than atat1µm 1 /xmsince since'ha) absorption will ftcoincreases increasesand andGex aex decreases, both leading leading to an increase in Icr. Icr The decreases, The maximum experiexperimental irradiance irradiance used j32 from the 0.53 0.53 jum mental used to to extract extract ß2 from the µm data was, was, therefore, increased increased to to22 GGW/cm2 . The therefore, W / cm2. Theabove aboveexperimental experimental considignoring free free-carrier calculating the erations justify ignoring -carrier absorption absorption in calculating transmitted irradiance irradiance[Eq. [Eq.(5)].13 (5)]. 13 transmitted The spatial beam beam profiles profiles in both both the the horizontal The horizontal and and vertical vertical directions were pinhole at the the directions were determined determined by by scanning scanning aa 25 25 /xm µm pinhole of the sample. sample. The The beam beam size size was adjusted at the sample sample by by position of pairs of of collimating collimating lenses. lenses. In all, four different spot sizes from using pairs 1.5 mm (FWHM) were used for the 1.06 1.06 pm /zm data. data. At At 0.53 0.53 0.5 mm to 1.5 (FWHM) were jum size used µm the beam size used was was 0.5 0.5 mm. mm. In In addition, addition, the beam profiles that there there were were no no hot hot spots, were monitored on a vidicon to ensure that spots, or shot shot-to-shot spurious reflections, or -to -shot beam beamwidth widthfluctuations. fluctuations. Figure Figure ^m beam of FWHM 33 shows shows aa representative representative pinhole pinhole scan scan for for aa 1.06 1.06µm 1.50mm. 1.50 mm. The incident energy energy was continuously varied varied using using aa stepping stepping-motor-controlled half-wave motorcontrolled rotating half -wave plate plate in in combination with with aa fixed polarizer. on the the sample sample fixed polarizer. This This apparatus kept the polarization on fixed and introduced no no measurable measurable beam beam walk walk with rotation rotation angle. angle. Previous experience experience indicates indicates that that other alternatives, alternatives, e.g., e.g., rotating rotating calcite polarizers, may cause the beam to walk across the sample and across the energyenergy-monitoring monitoring detectors. The choice of detectors, detectors, as as well well as as the the detection detection geometry, geometry, was was also critical. As discussed discussed in Sec. 5, phase also determined determined to to be critical. in Sec. 5, the phase aberrations introduced introduced on onthe thebeam beambybythe thetwo two-photon-generated -photon -generated free carriers cause considerable defocusing so that the beam profile at the detector detector varies with incident irradiance. Figure Figure 44 shows an example of this this defocusing defocusing as as observed in the near field behind a sample of CdTe. CdTe. As As the the irradiance is is increased increased the the beam beam broadens broadens and breaks x__. 7 0 1 POSITION(mm) Fig. 3. A pinhole pinhole beam beam scan scan showing a best best fit Gaussian Gaussian (solid Fig. 3. A showing a (solid line) line) of of FWHM 1.50mm. FWHM 1.50 mm. POSITION Vidiconscans scans(equivalent (equivalenttotoaapinhole pinholescan) scan) in in the thenear near field Fig. 4. Vidicon field of of the the /urn beam beam transmitted transmitted through through aa polycrystalline polycrystalline sample sample of of CdTe 1.06 µm CdTe showing the defocusing for for increasing increasing irradiance. irradiance. as isis characteristic characteristicfor forselfself-defocusing. 17 Thus, non-up, as defocusing.17 Thus, any spatial non uniformities in the detector response can lead to errors. Indeed, care be exercised exercised to to ensure ensure that that"external" "external"selfself-action 18 does must be action 18 does not result an occurrence result in in overfilling overfillingthe thetransmission transmissiondetector detector -an occurrence that that could result result in an overestimate overestimate of ß2 /32 (Ref. (Ref. 19) 19) or could could result result in in could discussed in optical limiting, as discussed in Sec. Sec. 6. 6. We We found, found, however, however, that by using large area detectors (1 (1 cm2) cm2) with withaameasured measured spatial spatial uniformity uniformity of better than than 10 10% %and and placing placing them them as as close close as as possible possible to to the the sample sample (3 cm), (3 cm), these these effects effects were wereeliminated. eliminated. The The detectors detectors were were also also deterdetermined to be be linear linear over over their their range range of of use use and andwere wereabsolutely absolutely mined calibrated with respect to a pyroelectric energy energy monitor* monitor* that was in absorbing-type turn checked against two others. others. In addition, absorbing -type neutral neutral density filters placed in front of of these detectors detectors were were checked to to have have transmission over over a range at least a factor linear transmission factor of 10 10greater greater than than the range used in these these experiments. experiments. Filters were never used used to to attenuate attenuate the beam prior to the sample. In addition, spike spike filters transmitting transmitting only 1.06 1.06 pun µm (0.53 (0.53 /zm) µm) were were placed placed directly directly in in front front of the detectors reduce optical noise noise from the flash flash lamps. lamps. to reduce *Gentec ED-100, *Gentec ED -100,Gentec GentecInc., Inc., Ste-Foy, Ste -Foy,Quebec, Quebec,Canada Canada 6167 OPTICALENGINEERING ENGINEERING // July/August Vol. 24 24 No. 616 / OPTICAL July /August 1985 // Vol. No. 4 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx PHOTON ABSORPTION, NONLINEAR NONLINEAR REFRACTION, REFRACTION, AND OPTICAL OPTICAL LIMITING LIMITING IN IN SEMICONDUCTORS SEMICONDUCTORS TWO PHOTON TABLE I.I. Material Parameters Parameters and Materials Studied Studied and Two-Photon Two- PhotonAbsorption Absorption Coefficients Coefficients of the Materials Two-photon /urn Two -photon absorption: absorption: AA == 1.06 µm Material Form(a > Form(a) ZnTe< c) ZnTelc) CdSe<c> CdSe(c) w W CdTe< d> CdTe(d) Z Z CdTe< d> CdTe(d) Zp Zp CdS0. 5 Se05< c> CdSo5$eo5(c) CdS0. 25 Se075<c> CdS025Seo(c) w W w W GaAs< e> GaAs(e) Z Z Z Z n(X) n(a) 2.79< c> 2.79(c) 2.56< c> 2.56ík) 2.84< c> 2.84(c) Eg (eV) Eg(eV) 2.26< j) 2.26(') 11.74))) .74<J> Ep (eV)< b> Ep(eV)Ibl ß2heor(cm/GW) /32 theor (cm/GW) /?fxp (cm/GW) ß2xP(cm /GW) Eb/Eg 19.1 19.1 0.004(o) 0.004(°) 21 21 22 25.1 25.1 15 15 25.1 25.1 4.5 0.89 11.44ík) .44< c> 11.44(k) .44<c> 20.7 0.007(i) 0.0071') 0.003(o) 0.003(°) 20.7 0.003(0) 0.003101 1.93< k> 1.93(k) 1.78< k> 1.78)k) 21 21 0.01 0(l) 0.01011) 10 12.1 12.1 2.51<'> 2.51(1) 21 21 15 15 17.7 3.43(i) 3.431') 1.42< 1.42(')j > 25.7 0.008W 0.008(1) 0.003(0) 0.003 °1 23 19.7 j8fxP(cm/GW) ßZxp(cm/GW) ßZheor(cm/GW) 02theor (cm/GW) 2.84<c> 2.84(k) 2.45<"> 2.451) Two-photon X = 0.53 0.53 µm , Two -photon absorption: X Form(a Form(a) n(X) n (X) ZnS<f> ZnS(t) ZnS<f> ZnS1t) Zp(clear) Zp(yellow) Zp(yellow) 2.40< 2.400)j) 2.40(i) 2.4011) ZnSe<9> ZnSe1g1 Zp CdS<c> CdS(c) ZnO' h> Zn01h1 W W Material 2f>o> 215w== 2.34 2.34 eV Eg (eV) Eg(eV) 18 18 18.6 2fio> = 4.68 eV eV 2tiw E(eV) Eb/Eg 3.66< 3.66(')j > 20.4 0.01 0(i) 0.010(') 2.0 20.4 24.2 0.01 Of'* 0.0101'1 0.008(0) 0.00810) 3.5 3.5 2.70< j > 2.70111 3.66< 3.66(')j > 2.67< 2.67(')j > 2.60< j > 2.601i1 2.42< c> 2.420) 21 21 0.01 2(i) 0.012)i) 5.5 5.5 2.10 2.10 4.27 4.87 2.05< j) 2.050) 3.20< n) 3.20(n) 21 21 0.020< 0.02d i)) 5.0 4.77 (a) ZZ == zinc polycrystalline (a) zinc blende; blende;W W == wurtzite; wurtzite; p = polycrystalline (b) Values the (b) Values taken taken from from Ref. Ref. 20. 20. For For values values not not listed in this Reference, the 21 eV eV was assumed. assumed. value of 21 (c) Cleveland (c) Cleveland Crystals, Crystals, Euclid, Ohio (d) II-VI, Pa. (d) II -VI, Inc., Inc., Saxonburg, Saxonburg, Pa. (e) Morgan Tex. (e) Morgan Semiconductors, Garland, Tex. (f) (f) CVD CVD Inc., Inc., Woburn, Woburn, Mass. (g) (g) Raytheon Raytheon Co., Co., Bedford, Bedford, Mass. 3.2. Data Data 3.2. Table I lists the 10 10 samples used in these experiments. In In all, all, eight eight different different materials materials having having either either aa zinc zinc blende blende or or wurtzite wurtzite structure structure were investigated. All of the samples were II II-VI -VI materials materials except for GaAs, which is a III III-V sample used used (0.5 (0.5 cm) cm) -V material. material. The thickest sample was was over over 100 100 times timesthinner thinner than than the the confocal confocal beam beam parameter (the Rayleigh distance) used used for this study. study. Experiments were performed on each single single crystal crystal sample sample for for two two orthogonal orthogonal directions of linear polarization. Within our experimental experimental accuracy, accuracy, no polarization. Within no anisotropy in the measured values /32 was was observed. observed. A 15% 15% variation was was the measured values for for ß2 reported in Ref. 26 26 for for room room temperature temperature CdTe; however, the optical pulse given. In addition, the the absolute absolute values values of of the the pulse width width was was not given. 2PA coefficients coefficients reported reported there there were were an an order of magnitude larger than those we we measured. measured. This This may may indicate indicate the the dominance dominance of carrier carrier absorption for long long pulses. pulses. In In ZnTe ZnTeand andthe thesingle single-crystal absorption for -crystal CdTe, the light light propagation propagation direction k was in the (110) (110) direction, direction, and in GaAs, k was in the (111) direction. InCdSe, CdSe,CdS0 CdS05Sep 5 Se05,5 ,and andCdS0.25Se0.75, CdS0 25 Se0 75 , direction. In k was parallel to the the cc-axis, -axis, while while in in CdS CdS and and ZnO, k was perpendicular to the c-axis. c -axis.Examples Examples of ofdata data used used to to extract extract the 2PA coefficient are shown shown in Figs. Figs. 5(a) 5(b) for 1.06 1.06 /urn /urn, respecrespec5(a) and 5(b) µm and 0.53 µm, tively. the average average of of five five laser laser firings. firings. The solid solid tively. Each Each data point isis the line W in 18 cm/ cm/ G GW line in in Fig. Fig. 5(a) 5(a) isis aa fit fit for for CdSe CdSe using using aa — = 0 and /32 ß2 = 18 Eq. (12). (12). The The solid solid line line in in Fig. Fig. 5(b) 5(b) isisaasimilar similarfit fitfor forZnSe ZnSeusing usinga.a == 0.5 cm" and ß2 /32 = 5.5 cm/ GW 0.5 cm I1 and GW at 0.53 0.53 jum. pm. In all samples the linear absorption value was was unimportant unimportant in in the the absorption was was small, small, and and its its value determination samples of ZnS ZnS the the scattering scattering was was In both samples ß2.. In determination of /32 significant, significant, and and this this loss loss mechanism mechanism was was included included in in the model model as Inthe thelatter lattercase, case,the theeffect effect of ofthe the choice choice of of aa on on ß2 /?2 linear absorption. absorption. In was less than than 10 10%. %. these measurements measurements of the 2PA 2PA coefficients coefficients are The results of these given given in in the the next next to to the the last last column column of Table Table I.I. The The absolute absolute error bars on the values of ft2 ±40%. %. The The relative relative error error 02 are estimated to be ±40 bars of ß2 /32 determined from one sample to the next are considerably better, as observed by measuring measuring all the samples in sequence for each spot size and ZnS (yellow) listed listed in spot size and each each pulse pulse width. width. For example, ZnS(yellow) 5.5 5.5 Chemetals, Plainview, Plainview, N.Y. N.Y. (h) Atomergic Chemetals, (i) Ref. Ref. 21 21 (i) Ref. 22 (j) Ref. Ref. 23 (k) Ref. (l)These values obtained uncton of composi (I)These obtained by by linear linear extrapolation extrapolation as aa ffuncton composition between the known known values values for forCdS CdS and and CdSe. CdSe. See Ref. Ref. 24. 24. (n) (n) Ref. Ref. 25 (o) Ref. 4 Table II always (t2 tnan We conservatively conservatively always had had aa larger ß2 than ZnS(clear). We estimate these these relative relative error error bars, which which are are important in determindependence of ofß2 /32 as discussed in the next section, ing the parametric parametric dependence be ±25 ±25%. to be %. second set of of experiments experiments (to be be discussed discussed in Sec. Sec. 5) we we In the second detector in in Fig. Fig. 11 by either a vidicon tube replaced the transmission detector interfaced with with an optical multichannel analyzer or a 25 25 pm µm pinhole pinhole placed at various radial positions and longitudinal distances from the We then monitored the spatial beam beam profile profile of of the transmittransmitsample. We vidicon, or we we monitored the pinhole pinhole transmistransmisted beam using the vidicon, irradiance. In InSec. Sec. 55 we we describe the use sion as aa function function of incident irradiance. (17) to theoretically fit both of of these these results. results. of Eq. (17) 4. COMPARISON OF ß2 02 VALUES TO THEORY 4. COMPARISON 4.1. Two-photon 4.1. Two -photon absorption theory We comparisons in in this this section. section. We We first first corn com-We make make three separate comparisons pare experimentally determined pare our experimentally determined /32 ß2 values values with with the the theory theory of of Refs. 3 3-5 Excellent agreement agreement isis Refs. -5 using using aa parabolic parabolic band structure. Excellent found for all materials except ZnTe. We then use the nonparabolic theory and again find good agreement agreement for for all all materials materials except except ZnTe. ZnTe. As shown by Weiler,4 Weiler,4 the differences between As shown the differences between the the parabolic parabolic and nonparabolic theories are minor so so that this this fit fit is expected. expected. We We then then include exciton correction correction factors, factors, as as given given by by Lee Lee and and Fan27 Fan 27 and as calculated when these these are included, included, calculated by by Weiler,4 Weiler,4and and we we find find that that when ZnTe nearly fits fits the dependencies dependencies shown shown by by the the other materials. As. Eg, and and As,stated statedininRef. Ref.5,5,the theparametric parametric dependence dependence of /32 ß2 on n, Eg, Ep wasfirst firstexplicitly explicitly pointed pointedout outby byPidgeon Pidgeonetetal.,3 al.,3 although although itit was E was present in the calculations of of Basov Basov et aí.28 al. 28 The The band band structure and transitionscheme schemeused usedby by Pidgeon Pidgeonetetal.3 al. 3 is shown in Fig. 6. transition 6. Calculations using this scheme have parabolic and tions using this scheme have been been performed performed for for parabolic nonparabolic bands. They found ß2 = 4Tre4 v m c2 ff; f ( 2flw ) Eg B = 53.8 2 n E8 /2fto)\ 2flcu f ~F~ ' Eg / \ ^g (I8) (18) OPTICAL / July/August 1985 / Vol. 617 OPTICALENGINEERING ENGINEERING / July /August 1985 / Vol.24 24No. No.4 4/ / 617 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx VAN STRYLAND, VAN STRYLAND,VANHERZEELE, VANHERZEELE,WOODALL, WOODALL,SOILEAU, SOILEAU,SMIRL, SMIRL,GUHA, GUHA, BOGGESS BOGGESS 3.2 2.4 116 ZnSe 0.53/am i- >k 1-6 (a) 21Ko 0.8 0.0 0.0 0.4 0.8 1.2 1.6 2.0 g - Zhu IRRADIANCE (GW/cm 2 ) 2.5i————i————i————i————i——— 2.0 Fig. 6. Band structure used used in Refs. Refs. 33 and and 44 to tocalculate calculatetwotwo-photon Fig. 6. Band photon absorption coefficients. 1.5 CdSe 1.06/um (b) As shown in Ref. Ref. 4, 4, the the differences differences between between f for for A«Eg and As shown 0«E and A»Eg, where0 is A is split-off energyshown shown in in Fig. Fig. 6, 6, are are small. small. A» E , where thethesplit -off energy is The expression expressionforforA»Eg 0» Eg is 1.0 fnp (x),A»Eg fnp(x), 0Eg =- 16 0.0 0.0 0.1 0.2 0.3 0.5 0.4 /x" IRRADIANCE (GW cm2 ) where Ep andEEg andß2/£>inincm/ cm/GW the last last expression. expression. where E and areareinineVeVand GW ininthe Here, m is is the the electron electron mass, e isis the the electron electron charge, c is is the the speed speed of of light refractive index. index. The The values values of ofEg, Eg , E0, Ep , light in in vacuum, and nn is the refractive and n for each material are listed in Table I.I. It is is important to note note that for for both both parabolic parabolic and and nonparabolic nonparabolic bands, bands, the the parametric parametric dependencies on n, Eg Ep predicted by the theories are the same. E ,, and Ep same. In addition, addition,asaspointed poinfed out in in Ref. Ref. 55 these these dependencies dependencies are indeinde In pendent of crystal structure. structure. The The differences differences lie lie in the function f and the of fico the ratio ratio of tut) to to Eg Eg(i.e., (i.e.,which whichstates statesare are optically optically coupled). coupled). Weiler Weiler44 corrected Ref. 33 and corrected an error in the calculation of f in Ref. and obtained the following following expression expression using using parabolic bands: = - 4 + 29 /2 (4+- [(x z = 96.9 [F2(x)] 12 . (19) (19) i TT 3 2x-1 7 + 45 fnp (x),A«Eg =y 3 3/x~ X X 3 2x x •+ ( (x + I)3 / 2 A x 3x5 5 — (x4 (x4 + 2x2 2x2 + -f 6) 6) )3/2 . (20) (20) -1)2 < x5 ( 55, x4+2x2+6/I — x2 4X + — x 16 X 16 .. (21) (21) results of of these these calculations calculations of ofthe the excitonic excitonicenhancement enhancementgeX ggx of of/^ results ß2 as as a function of of Co/ fico/Eg various values values of ofe.e. Eg for various 4.2. Comparison Comparison to theory theory 4.2. /(/ETTF2) Eq. Figure 88 shows shows aa log-log log -logplot plotofofp2ß2scaled scaledbybyn2n2/ ( P F2) [see Eq. (19) F2] as Eg . The is aa least (19) for for F2] as aa function function of Eg. The solid line is least squares squares fit to the data (excluding ZnTe) for a line line having havingaaslope slopeof of—3 -3 to account Eg 3 dependence of ß2. fa. Clearly, dependences for the Eg3 Clearly, the parametric dependences using this fit the the data datavery verywell well except except for for ZnTe. ZnTe. using this parabolic theory fit This single parameter fit yields yields aa 2PA 2PA coefficient coefficient given given by by the This single parameter fit the follow following ing equation: (3.1±0.5)X103 ß2 = (3.1 t0.5)X 103 fnp(x), 0«Eg = 5(x x3 to values values of of ß2 /32 have been predicted by Lee and Exciton corrections to Fan. 27 Weiler 4 has ratio ee Fan.27 Weiler4 hasevaluated evaluated these these corrections corrections in in terms terms of the the ratio binding energy energy Eb Eb to to the theband band-gap energy E2. Eg. These These of the exciton binding -gap energy ratios are are listed listed in in Table I.I. We We reproduce reproduce in in Fig. Fig. 77 (from (from Rif'. ReT. 4) 4) the the where where F2 F2(x) (x)isisthe thesame samefunction functiondefined definedininRef. Ref.5.5.For Fornonparabolic nonparabolic Weiler 4 found bands, Weiler4 found for 0A«EP << Eg 32 (x - I)3 / 22 3 _6_ +1)3/2 I)3 / 2 6 (x + X Fig. 5. Inverse transmission transmission versus (a) CdSe CdSe at Fig. 5. Inverse versus incident incident irradiance irradiance for for (a) at 1.06 /*m /urn. The solid lines are Eq. (12). µm and (b) (b) ZnSe ZnSe at at 0.53 0.53 pm. are fits fits using Eq. f(x) = (x --1)3/2 I)3 / 2 (x » 0.5 P Ma)) (-E /2fico\ n2 E3 n2E3 (22) (22) where again areinineV eV and andß2 /32 isis in in cm cm/GW. The values values where again Ep Ep and and Eg E are /GW. The predicted predicted from from Eq. Eq. (22) (2Z)are are listed listedininthe the last last column column of of Table Table II for for each material. material. The The value value of ofthe theconstant constant[3.1 [3.1X103 Eq. (22)] (22)] each X 103 ,in in Eq. predicted by theory theory from from Eqs. Eqs. (18) (18) and and(19) (19) isis 5.21 5.21 X X103 predicted by 103,, so sothat that the the absolute values values of of the experimentally experimentally determined determined ß2 /32 values values are, are, on on 6187 OPTICALENGINEERING ENGINEERING // July/August Vol. 24 No. No. 4 618 / OPTICAL July /August 1985 1985 // Vol. 4 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx SEMICONDUCTORS IN SEMICONDUCTORS REFRACTION, AND PHOTON ABSORPTION, NONLINEAR REFRACTION, AND OPTICAL LIMITING IN TWO PHOTON 10' I ' I -e-0.01 E =0.01 -0.008 0.008 -0.006 0.006 -0.004 0.004 -0.002 0.002 -0.0005 0.0005 I Saga a)s 10° 100 0.5 I 0.6 j_ j_ 0.7 0.1 0.8 I 0.9 1.0 10 GAP ENERGY/ENERGY PHOTON PHOTON ENERGY /ENERGY GAP various fkw/Eg of hw gex as aa function factor Fig. 7. Exciton g8% function of /Eg for for various enhancementfactor Excitonenhancement band-gap values values of of the the exciton exciton binding energy Eb to the band -gap energy energy Eg (repro4). Ref. 4). duced with duced with permission from Ref. coefficients absorption two-photon scaled thescaled ofthe plot of Fig. 9. A log -log plot twophoton absorption coefficients log-log gap. The solid energy gap. versusenergy band structure structure (A«Eg) (0«E9) versus solid nonparabolic band for nonparabolic ZnTe). (omittingZnTe). line is aa least least squares squaresfit fit of of the the data data to to aa line line of of slope slope-3 -3 (omitting The The dashed dashed line line isisthe the theory theory of of Ref. Ref. 4. GaAs 1000 1000 -CdTe CdSe - eó CI ZnTe CdS.25$e.75 C2 CdS 5Se 5 100 ZnSe CdS 100 1.06/um 1.061.im--r -0.53µm 1.5 2.0 2.5 3.0 3.5 4.0 1.5 2.0 2.5 3.0 ZnS. Zñ0 3.5 4.0 4 0 Eg(ev) Eg)ev) two-photon scaled twothe scaled log-log Fig. 8. Fig. 8. A log -log plot of the photon absorption coefficient versus versus energy energy gap gap assuming assuming parabolic parabolic band band structure. structure. The solid line is aa ZnTe). The x's (omittingZnTe). 3 (omitting least east squares squaresfit fit of of the the data data to to aa line lineof ofslope slope -3 Ref. 13. Data Data to CdTe, CdSe, and ZnTe GaAs, CdTe, for GaAs, shown for ZnTe are are data data from from Ref. and to the line were taken with 11 /urn pm light and the left of the vertical dotted line M light. right with 0.5 µm 1.7. of 1.7. the the average, average, low low by by aa factor factor of values with /^ values determined ß2 If we If we now now compare compare the the experimentally experimentally determined with results the results obtain the we obtain (20)], we Eq.(20)], the nonparabolic nonparabolic theory theory[A«E [0 «Eg, Eq. the is shown shown in in Fig. Fig. 9. 9. While While the the overall overall tit Tittotothe theparametric parametric dependence dependence is is aa line is solid line the solid (again the case (again parabolic case the parabolic for the as for good as as good quite as not not quite excluding ZnTe), squares fit least squares least fit to to aa line line of of slope slope—3, -3, excluding ZnTe), the the absolute absolute this by this predicted by lower than 26% lower only 26% average, only the average, values are, values are, on the than predicted (20). and (20). (18) and Eqs. (18) by Eqs. predicted by curve predicted the curve line isis the dotted line The dotted theory. The theory. an gives an similar plot A similar plot (not (not shown) shown)using usingEq. Eq.(21) (21)for forA»Eg 0 »Eg gives theory between theory difference between the difference case the this case in this except in fit, except identical fit, almost identical almost 3.5%. only3.5 reducedtotoonly dataisis reduced the data of the fit of squares fit least squares and aa least and %. The The actual actual although theories, although two theories, the two between the in between lies in materials lies these materials for these case for case the in the scatterin forscatter responsiblefor partiallyresponsible maybebepartially andmay Eg,and closer to closer to A« A «Eg, /32). in ß2). change in 23% change to 23% (up to data data (up in given in as given excitons, as /32 due If If we we now now include include corrections corrections to to ß2 due to to excitons, the used the have used we have Here, we 10. Here, Fig. 10. of Fig. curve of the curve we obtain the 7, we Fig. Fig. 7, g^. with gex. scaling with F2 ), except (i.e., F2), theory (i.e., parabolic parabolic theory except for for the the additional additional scaling bands. 27 parabolicbands.27 forparabolic onlyfor calculatedonly beencalculated hasbeen factorhas This correction factor This correction we see What What we see in in Fig. Fig. 10 10isisthat thatmost mostof ofthe thediscrepancy discrepancy between between theory theory by removed by been removed has been 4.3)^ has (gex ==4.3)4 ZnTe(gex forZnTe observedfor and experiment observed and experiment Fig. 10. A log-log log -logplot plot of of the the scaled scaledtwo-photon two- photonabsorption absorption coefficients coefficients (gex ) for enhancement (gex) exciton enhancement including exciton for parabolic band structure structure versus energy gap. the to the including including excitonic excitonic enhancement. enhancement. However, However, the the overall overall fit fit to is fit) is squares fit) least squares lineisis aa least parametric dependence straightline the straight (againthe dependence(again parametric in accuracy in of accuracy lack of our lack somewhat somewhat worse. worse. This This may may be be attributed attributed to to our is of ggx is materials, gex calculating gex. Note that that for for most of the the order order of of 2, 2, most materials, ggx . Note calculating 7). (see Fig. 7). well above the gap (see even when the the coupled coupled states are well even when Consequently, Consequently, when when excitonic excitonic enhancement enhancement is is included included for for parabolic parabolic of 3.3 /32 values bands, the the absolute absolute ß2 values predicted predicted by by theory theory are are aa factor factor of 3.3 bands, measured. those measured. larger larger than than those consider is considerZnTe is for ZnTe enhancement for exciton enhancement the exciton that the reason that The reason two jum, two 1.06 µm, materials isis that the other materials ably ably larger larger than than for for the that at 1.06 excitonic where excitonic gap, where 3.5% above the gap, only 3.5% couple states only photons photons couple out, point out, We should 7. We Fig. 7. in Fig. shown in as shown be greatest, effects should effects should be greatest, as should point however, however, that that we we have have only only one one material material for for which which the the coupling coupling isis larger material are this material for this bars for this this close close to to the the gap, gap, and and the the error error bars are larger use to use us to allowing us easily, allowing damagedeasily, ZnTe damaged materials. ZnTe for the than than for the other other materials. fitting for fitting small spot aa small spot size size only only and and limiting limiting the the range range of of irradiances irradiances for overlap is overlap where there is 8), where Fig. 8), in Fig. 02.. On On the the other hand (as shown shown in /32 Bechtel and by Bechtel taken by between between the the data data taken taken here here and and the the data taken /?2 for large ß2 13 the Smith, Smith,13 theagreement agreement isisexcellent. excellent. They They also also obtained obtained aa large for ZnTe. ZnTe. 9, 8, 9, Figs. 8, in Figs. shown in and shown Table II and in Table listed in points listed data points Two Two other other data an gave an sample gave CdTe sample polycrystalline CdTe The polycrystalline comment. The require comment. 10 require and 10 and is single-crystal the single lower than the )32 lower experimental ß2 experimental -crystal CdTe CdTe sample. sample. It It is 619 / / 619 44 / Vol. 1985 / July/August OPTICAL OPTICALENGINEERING ENGINEERING / July /August 1985 / Vol.2424No. No. Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx VAN STRYLAND, STRYLAND, VANHERZEELE, VANHERZEELE,WOODALL, WOODALL, SOILEAU, SOILEAU, SMIRL, GUHA, BOGGESS BOGGESS designed as as aa 10.6 10.6 µm )um optical opticalwindow windowand andisisdoped dopedwith with1017 10 l7 cm cm~3 of designed -3 of indium. is not understood understood at present indium. ItIt is present ifif or or how how these these impurities impurities lower the the measured measured /32. /32 . The sample is is aa chemical chemical vapor vapor lower The ZnS(yellow) ZnS(yellow) sample grown sample sample that has has aa yellow yellow appearance caused caused by by deposition grown crystal lattice lattice imperfections imperfections that annealed out by by aa special special crystal that can be annealed heat process. The The ZnS(clear) ZnS(clear) is is the the same same starting heat treatment treatment process. starting material material that has has undergone undergone this this heat heat treatment. treatment. ItIt appears appears water water clear, clear, and and its its that linear transmission transmission cutoff cutoffisis shifted shifted into into the the UV. U V. The from linear The indication indication from the data data presented presented here here is is that not only only has has the the linear linear absorption absorption in in the that not the visible visible and near-UV but the the two two-photon absorp the and near -UV been been reduced, reduced, but -photon absorption at tion at 0.5 0.5 pm jum has has also also been been reduced reduced by by this this heat heat treatment. treatment. That That is, is, defects may may be be contributing to02. /32 . This This may may also also be be true true for for the the ZnTe ZnTe defects contributing to sample since it it damaged damaged easily. easily. sample at at 11 /urn µm since (b) 5. SELF -REFRACTION 5. SELF-REFRACTION In section we beam propagation propagation behind behind In this this section we present present results results for for the the beam the sample, sample, given given the irradiance and and phase phase distributions distributionsofofEqs. Eqs.(10) (10) the the irradiance (j = 2) and (14). Equation Equation(17) (17) describes describes the thefluence fluence atatany anypoint point(z(z,, r) r) (j = 2) and (14). behind the the sample. sample. We We present present data data here here using using aa beam beam of ofwidth width 1.70 1.70 behind mm (FWHM) incidenton on the theCdSe CdSesample samplelisted listed in in Table Table I,I, which which is is mm (FWHM) incident placed near near the the beam beam waist. waist. Thus, Thus, c(o, 4>(o, r, in Eq. Eq. placed r , t)t) isistaken taken to to be be zero zero in (14). The signal was was monitored monitoredatatdistances distancesofof0.5 0.5 m m and and (14). The transmitted transmitted signal m behind behind the the sample, sample, both both of ofwhich which are are near near-field regions. Figure Figure 22 m -field regions. 11 illustrates 11 illustrates the the change change in in the the beam beam profile profilewith withirradiance irradianceatat zz == 0.5 pulses) as as displayed displayed on on the the vidicon. vidicon. This This 0.5 m m (using (using 92 92 ps ps FWHM FWHM pulses) scan is is aa narrow slice through the center center of of the the beam beam that that isis equivaequiva scan narrow slice through the lent to to aa pinhole pinhole scan. scan. The The dashed dashed line line isis aa fit fit to to Eq. Eq. (17) (17) as lent as aa function function ofrfor/32 18cm/GW,P of r for ß2 = = 18 cm/ G W, P== 3,andy 3, and y = = 00(i.e.,n2=0). (i.e., n2 = 0). We Wefind find that that ifif present, effects of bound electronic electronic contributions non present, the the effects of bound contributions to to the nonlinear refractive index overshadowed by by the photogenerated photogenerated linear refractive index are overshadowed carrier effects. effects. This This is is confirmed by our other measurements, measurements, which which carrier confirmed by our other monitored fluence transmitted pinhole as function of of monitored the the fluence transmitted by by aa pinhole as a function irradiance. irradiance. Using aa 25 we Using 25 /um pm diameter diameter pinhole pinhole in in front front of aa photodetector, we measured the the on on-axis 0) fluence signal as as aa measured -axis (r(r = = 0) fluence of of the the transmitted transmitted signal function of the the peak peak on on-axis irradiance for for two two pulse pulse widths widths function of -axis input input irradiance and distances. The results are shown in Figs. 12 12 and 13. 13. To and two distances. The results are shown in Figs. further verify the the validity of the the irradiance irradiance dependence dependence of of further verify validity of the theory, theory, the the transmitted transmitted fluence fluence at at an an off off-axis was also measured. This This the -axis point point was also measured. is shown shown in in Fig. Fig. 14. 14. is The fits in in Figs. Figs. 12, 12, 13, 13, and 14 were the The theoretical theoretical fits and 14 were obtained obtained from from the numerical evaluation evaluation of of FF given given in in Eq. Eq. (17). (17). Other Other parameters parameters(see (see numerical Table 1) I) used were aa =— 0.2 0.2 cm cm"-11 and and meh meh = = 0.104 0.104 Table used in in the the calculation calculation were m. 22 The by two two-photon M.22 Thetotal totaldensity density NN of of charge charge carriers carriers generated generated by -photon absorption on on-axis can be be calculated by integration integration of of Eq. Eq. (8). (8). At At aa absorption -axis can calculated by irradiance of of 11 GW GW/cm2 peak value value of of N N peak input irradiance /cm2,, we we obtain obtain a peak evaluated at at the theinput inputplane planeofofthe thesample sampletotobebe2 X 2X10 cnT"3 We evaluated 101818 cm -3.. We find that the best best agreement agreement between between the the experiments find that the the theory theory and and the experiments is obtained for PP ==3.5. 3.5. The The fits fits shown shown in in Figs. Figs. 12, 12, 13, 13, and 14 are is obtained for and 14 are for for values of 3.5 ±±1,1, with with the theexact exactvalue value of ofPP adjusted adjustedbetween between4.5 4.5 values of PP -=3.5 and 2.5 2.5 to to obtain obtain the the best best fit. fit. Using Using aa two two-level been and -level model, model, PP has been calculated to be be Egg E2 /(E| 29 For = 1.74 1.74 eV eV at room calculated to /(Eg-liV). -t2w2).29 ForCdSe, CdSe,Eg E= at room temperature, and and this mis formula formulapredicts predictsPP ==22 at at 11 µm. /um. The The peak peak phase phase temperature, change change undergone undergone by by the the beam, beam, calculated calculatedfrom fromEq. Eq. (14), (14), isis (for (for Io I0 = = 11 GW/cm2 ) A<£ which is is 1.3 1.3 wavelengths wavelengths distortion. We GW /cm2) 0(1)==—8.1, -8.1, which distortion. We have ignored ignored the the n2 n2 due bound electronic electronic effects effects in in these these calculacalcula have due to to bound tions. Even Even ifif this this nonlinearity nonlinearityfor forCdSe CdSewere wereasashigh tions. highasasthe then2 n2 of ofCS2 CS2 (i.e., 10 10~-11 n esu), the phase phase change change (i.e., esu),the the maximum maximum contribution contribution to the would be40n-Llwn2/ be407rLIcon2 /n0 for to I0 = = 11 GW/ GW/cm2 would no,, which which is0.2 is0.2 for cm2.. This This index index change also also would would be be aa self self-focusing not aa defocusing defocusing change -focusing effect effect and and not effect as the nonlinear nonlinear refraction refractionobserved observedinin CdSe CdSeisis effect as observed. observed. Thus, Thus, the approximately 40 40 times times larger larger than than in in CS2 CS2 at at this this irradiance. irradiance. Higher Higher approximately irradiances give give rapidly increasing increasing values values of of defocusing defocusing since since the the irradiances nonlinearity isis induced of two two increase increase in in nonlinearity induced by by 2PA 2PA (i.e., (i.e., a factor of irradiance gives gives nearly larger phase phase distortion). distortion). irradiance nearly aa factor factor of of four four larger OPTICAL LIMITER LIMITER 6. OPTICAL In section we describe describe a nonlinear nonlinear optical optical device device (an optical optical In this section power limiter)1O limiter) 10 that power thatutilizes utilizesboth both two-photon two -photon absorption, absorption, as as dis- (a) -..- i --- I -202 -2 2 0 TRANSVERSE DISTANCE (mm) TRANSVERSE DISTANCE (mm] 4 Fig. scan of beam beam transmitted through the polycrystalline polycrystalline Fig. 11. 11. Vidicon scan sample sample of of CdSe CdSe at at (a) (a)high high irradiance irradiance (1(1 GW/cm2 GW /cm2)) and and(b) (b)low low irradiance irradiance (0.3 GW GW/cm2 m behind behind the the sample sample (near (near field). The The /cm2)) at aa distance distance of 0.5 m pulse pulse width width used beam profiles profiles are normalized to used was 92 ps FWHM. The beam have have the the same same on-axis on -axisfluence. fluence. The The dashed dashed line line is is experimental experimental and the solid using Eq. Eq. (17). solid line is is aa theoretical theoretical fit fit using cussed Sees. 33 and 4, 4, and and the theassociated associated nonlinear nonlinear refraction refraction cussed in in Secs. discussed discussed in in Sec. Sec. 5. 5. This This completely completely passive passive device device has has aa high high trans transmission for it clamps at mission for low low input input irradiance irradiance (fluence), (fluence), but but it clamps the the output output at aa constant constant irradiance above aa predetermined irradiance (fluence) (fluence) above predetermined input. input. Such Such aa device restrict the the irradiance irradiance device can can be be used used as as aa protective protective element element to to restrict (fluence) of of aa pulse pulse incident incident upon upon sensitive sensitive optical optical components components or (fluence) or as as a regulator to smooth smooth optical optical transients. transients. This This device device is the optical analog of aa Zener Zener diode. diode. analog of Optical limiting by absorption in Optical limiting by nonlinear nonlinear absorption in semiconductors semiconductors was was proposed and and demonstrated demonstratedininthe thelate late1960s.30 1960s. 30 "-32 32 Moreover, proposed Moreover, non nonlinear linear refraction refraction combined combined with with spatial spatial filtering filtering has has been been used used to to demonstrateoptical opticallimiting limiting liquid33 - 34and andgas gas-filled cells. 35 Here Here demonstrate in in liquid -33.34 -filled cells.35 we demonstrate optical limiting limiting in in GaAs. GaAs. Below Below the the melting we demonstrate optical melting threshthresh old, what we we have have done differently isis to use not only old, what done differently to use only nonlinear nonlinear GaAs but but nonlinear nonlinear refraction refraction together together with with absorption (2PA) in GaAs spatial construct a more more effective effective device. device. Above Above the spatial filtering filtering to to construct melting we also also take take advantage advantageofofaasolid solid-to-liquid phase melting threshold, threshold, we -to- liquid phase transition, which affects affects the the reflectivity reflectivity and transmission. transition, which and transmission. The geometry geometry we we used limiting is is shown Fig. 15. 15. A The used for for optical optical limiting shown in in Fig. A single 1.06 ^m pulse was was focused focused to 100 µm /um single 40 40 ps ps (FWHM) (FWHM) 1.06 µm pulse to a 100 (FWHM)spot spotatatthe thesurface surfaceofofthe theGaAs GaAswith withaa465 465 mm mm focal focal length length (FWHM) lens beam was was collected collected and and collimated collimatedby byaa381 381 lens Lj. L1. The The transmitted transmitted beam mm mm focal focal length length lens lens L2 L2placed placed one one focal focal length length behind behind the the sample. sample. The recollimated recollimated beam beam then then passed passed through through aa 22 mm mm diameter diameter aperaper The ture placed placed one one focal focal length length beyond L2 and of aa beyond L2 and directly directly in in front of photodiode. sample used sample photodiode. The The sample used was was nearly nearly identical identical to to the sample listed in in Table Table I. listed I. The The limiting limiting capabilities capabilities of of the the device device are are shown shown by by the triangles in 16. At /itJ), the the device device the triangles in Fig. Fig. 16. At low low input input energies energies(<0.5 (<0.5 pi), response was linear response was linear and and was was consistent consistent with with the the 45% 45% linear linear transmis transmission the 73% 73% transmission of the the pinhole. pinhole. Above Above10 10 sion of of the the GaAs GaAs and and the transmission of /zJ input, input, the the output output energy energy was was essentially essentially clamped µJ clamped at at 11 )uJ. µJ. The device device continued continued to to limit limit for for input input energies energies greater greater than than 11 mJ. mJ. Over Over the full range of operation, operation, the the system system transmission transmission was was reduced reduced the full from 33% to input enerener from 33% to 0.1%. 0.1 %.Notice Noticethat thatregulation regulation continued continued for for input gies the GaAs GaAs single single-shot of 0.9 0.9 JJ/cm2 gies far far above above the -shot melting melting threshold threshold of /cm2 (indicated by the arrow arrow in in Fig. Fig. 16). I6). Above Above the the melting melting threshold, threshold, the the (indicated by the GaAs before each each firing firing so each pulse pulse irradiated irradiated GaAs was was translated translated before so that that each virgin virgin material. material. Below for melting, melting, as energy was Below the the irradiance irradiance required required for as the the input input energy was increased, the increased, 2PA. In the transmission transmission of of the the GaAs GaAs was was reduced reduced by by 2PA. In this this 620 // OPTICAL Vol. 24 4 620 OPTICAL ENGINEERING ENGINEERING//July/August July /August 1985 1985 // Vol. 24 No. No. 4 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx SEMICONDUCTORS IN SEMICONDUCTORS LIMITING IN OPTICAL LIMITING REFRACTION, AND NONLINEAR REFRACTION, AND OPTICAL TWO PHOTON ABSORPTION, NONLINEAR la) XX lb) 0 0.00 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 cm2 ) INCIDENT INCIDENT IRRADIANCE IRRADIANCE (GW (GW /cm2) 0.5 0.5 cm2 ] (GW /cm2) IRRADIANCE (GW INCIDENT IRRADIANCE INCIDENT irradiance at incident irradiance Fig. -axis fluence as aa function function of incident on-axis Transmittedon Fig. 12. Transmitted and (b) 43 ps pulses and FWHM pulses ps FWHM (a) 92 ps sample: (a) the sample: behind the distance of 0.5 m behind aa distance Eq. (17). pulses. The FWHM pulses. FWHM The solid solid lines lines are are numerically numerically calculated calculated from Eq. 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 i as a off-axis as measured at Fig. Fig. 14. 14. Transmitted fluence measured at aa point point 1.1 1.1 mm off-axis sample: the sample: behind the m behind 0.5 m of 0.5 distance of ataa distance irradianceat incidentirradiance ofincident function of function are lines are solid lines The solid pulses. The FWHM pulses. ps FWHM and (b) 43 ps pulses and (a) 92 ps FWHM pulses (a) Eq. (17). numerically numerically calculated calculated from Eq. f2 H A GaAs 1 2mm _L V L2 A Fig. Fig. 15. 15. Schematic of GaAs GaAs optical optical limiter. limiter. 10* 102 . »';i-/. . . f ..:'V 10' 101 • ••** 0.0 i i i 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 cm2 ] INCIDENT INCIDENT IRRADIANCE IRRADIANCE (GW (GW /cm2) irradiance at incident irradiance of incident on-axis Transmittedon Fig. 13. Transmitted -axis fluence as aa function function of and (b) 43 ps pulses and FWHM pulses ps FWHM 92 ps (a) 92 sample: (a) the sample: behind the m behind 2.0 m a distance of 2.0 Eq. (17). FWHM pulses. The solid solid lines lines are are numerically numerically calculated calculated from Eq. pulses. The i5 ir" " = 10o» A * lo-'i GaAs GaAs 10 -1 by the regime, regime, the the amplitude amplitude of of the the spatial spatial beam beam profile profile transmitted transmitted by the by distorted by was distorted phase was the phase solely by GaAs was was distorted distorted solely by 2PA, 2PA, and and the GaAs 2PA-generated the 2PAindex index changes changes associated associated with with the generated free free carriers, carriers, as as self-diffraction discussed discussed in in Sees. Secs. 22 and and 4. 4. The The selfdiffraction associated associated with with this this effective pinhole the effective reduced the phase and and amplitude amplitude distortion distortion reduced pinhole trans transphase reflectivity the reflectivity in the increase in the increase melting threshold, mission. Above Above the the melting threshold, the mission. GaAs the GaAs reduced the further reduced region further molten region absorptivity of the molten and and absorptivity the of the profile of amplitude profile distorted the amplitude transmission transmission and and further further distorted beam. transmitted transmitted beam. reflection to nonlinear reflection of 2PA The The contribution contribution of 2PA and and nonlinear to the the limiting limiting action action can can be be separated separated from fromthe thecontribution contribution of of nonlinear nonlinear refraction refraction by carefully collecting collecting all all of ofthe the transmitted transmitted energy energy with with the the pinhole pinhole by carefully Aperture No Aperture • No Aperture 2mm Aperture A 2mm ETH 10-2: 10- TH (o) 1 * i 40 40 ill 80 80 120 120 i i 160 160 200 200 240 'I 240 ENERGY ENERGYININ|/MJ) (µl) (circles) the 2 mm (triangles) and without response with Fig. 16. 16. Device with (triangles) without (circles) Device response Fig. aperture in place. ETH ETH represents the single-shot single -shot melting melting threshold. threshold. circles the circles by the shown by are shown results are These results 16. These Fig. 16. in Fig. shown in as shown removed, as removed, by dominated by is dominated Below ETH 16. Below Fig. 16. in in Fig. ETH,, the the nonlinear nonlinear transmission transmission is GW. 10 The cm/GW.IO 26 cm/ coefficientofof26 2PA coefficient by aa 2PA fit by be fit can be and can A and 2P 2PA The value value of of 621 / Vol. 1985 / July/August OPTICALENGINEERING ENGINEERING / July /August 1985 / Vol.2424No. No.4 4/ / 621 OPTICAL Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx VAN STRYLAND, STRYLAND, VANHERZEELE, BOGGESS VANHERZEELE,WOODALL, WOODALL, SOILEAU, SOILEAU, SMIRL, GUHA, BOGGESS 26 26 cm/GW cm/ GW reported reported in in Ref. Ref. 10 10 for for the the 2PA 2PA coefficient coefficient ß2 fa of of GaAs GaAs is is nearly 23 cm/ cm/GW nearly identical identical to to the value of 23 GW shown shown in in Table Table I and indicates the the confidence confidence in in the the values values of of ß2. /32 . These These measurements measurements of of indicates j32 on GaAs were were made made independently independently with with aa nearly nearly identical identical laser laser ß2 on GaAs system. Above Above ETH, ETH , the system. the problem problem becomes becomes much much more more complicated complicated is outside outside the scope scope of this paper. In In this this regime, regime, the central and is region pulse that arrives arrives after after melting melting is is initiated initiated isis heavily heavily region of of the the pulse attenuated and GaAs. In addition, and reflected reflected by the molten layer of GaAs. there is is considerable of material material for for fluences fluences more there considerable evaporation evaporation of more than than — 10% above above the the melting melting threshold. threshold. It It isis clear, clear, however, however, that the the -10% transition from from below below to to above above threshold threshold is is aa smooth smooth one. one. That That is, is, transition there is is no discontinuity in in limiter limiter response response at threshold. there no discontinuity at threshold. It is is interesting compare the the contributions contributionsofof2PA 2PAand andself self-dif It interesting to to compare -difinput energy energy of 80 fraction just below the melting threshold. threshold. At an input /uJ, 2PA 2PA acting acting alone alone has has reduced reducedthe thetransmission transmissionby byaafactor factorofoffive. five. µJ, On the other other hand, hand, the the combined combinedeffects effects of of 2PA 2PA and and selfself-diffraction On the diffraction have reduced 30. Thus, present have reduced the the transmission by a factor of 30. Thus, the present configuration is aa considerable configuration is considerable improvement improvement over over limiters limiters that that uti utilize 2PA exclusively. lize 2PA exclusively. We the present present switch switch with with the the Si Si device device We should should also also contrast the demonstrated in Ref. Ref. 36 36 that that utilizes utilizes indirect indirect absorption, absorption,free free-carrier demonstrated in -carrier absorption, self-diffraction, andsolid solid-to-liquid phase change change to to limit limit absorption, selfdiffraction, and -to- liquid phase energetic /zm. The The nonlinear nonlinear absorption absorption in in Si Si at at 11 µm jum is is energetic pulses pulsesatat 11µm. strictly fluence-dependent, is independent independent strictly fluence- dependent, and and the the device device operation operation is of pulse pulse width width for for pulse pulse widths widths shorter shorter than the carrier recombinarecombina tion time. time. The The present present device device isis considerably tion considerably more more complicated. complicated. The The nonlinear absorption, which which isis dominated dominated by by 2PA, 2PA,isisirradiance irradiance-nonlinear dependent, while while the arises from dependent, the nonlinear nonlinear refractive refractive index index that that arises from the the 2PA-generated time-integrated persists 2PA -generated free free carriers carriers is is aa time -integrated effect effect that that persists the duration duration of ofthe thecarrier carrierlifetime. lifetime. Nevertheless, Nevertheless, the the carriers carriers for the be generated generated without without aa sufficiently sufficiently intense intense pulse, pulse, which which in cannot be practice practice restricts restricts this this device deviceto to operation operation with with short short pulses. pulses. However, However, the the limiting limiting pulse pulse energy energy can can be be varied varied considerably considerably by by changing changing the the geometry geometry (e.g., (e.g., using using aa very very short short focal focal length length lens lens L1 Lj in in Fig. Fig. 15 15 will will lower lower the the limiting limiting energy). energy). An An advantage advantage of of the the present present device device (and (and 2PA-based limiters in in general) general) over over the the Si Si device device isis its its higher higher 2PA -based optical optical limiters linear advantage of of linear transmission transmission at at 11 /zm. µm. Another Another more more important important advantage 2PA-based offer. For For 2PA -basedlimiters limitersisisthe the broader broader bandpass bandpass that that they offer. example, the the GaAs GaAs device device should should function function for for wavelengths wavelengths between between example, approximately 0.9 0.9 and and 1.7 1.7 µm, /zm, where where 2PA 2PA is is the absorp approximately the dominant dominant absorption process. tion process. 7. CONCLUSION 7. CONCLUSION The material material parameter parameter dependence dependence found wide variety variety of The found for for aa wide semiconductors, as semiconductors, as discussed discussed in in Sec. Sec. 3, 3, allows allows us us to to predict, with reasonable confidence, confidence, the two-photon coefficient of reasonable the two -photon absorption absorption coefficient of other materials materials at at other otherwavelengths. wavelengths. This This includes, includes, for for example, example, other mixed ternary Thus, the the 2PA 2PA at at aa particular wavelength mixed ternary compounds. compounds. Thus, particular wavelength can be tailored tailored for for aa specific specific application. application. can be The fact fact that that this this scaling scaling fits fits the the data data so so well well implies implies that that all all of of the the The important important material material parameters parametershave have been been included included in in the the theory theory and and that other contributions contributionstotoß2 fa (e.g., (e.g., higher higher bands) bands) cause cause small small effects. that other effects. One possible deviation the predicted predicted scaling scaling is is the the effect effect of of One possible deviation from from the excitons. material (ZnTe) (ZnTe) that indicates indicates that such such excitons. We We have have one one material effects may If we we attempt to extend extend this this theory theory into into the the effects may be be important. important. If attempt to UV, the predictions predictions are are in in general general considerably considerably lower lower UV, we we find find that that the than in the however, aa general general trend trend can can be be found. If the the than in the experiments; experiments; however, found. If coupled are well well above gap, the the deviations deviations are are relatively relatively coupled states states are above the the gap, small, order of of aa factor of four four or or five. five. As As the get small, on on the the order factor of the coupled coupled states states get close exam close to to the the gap, gap, however, however, the the deviation deviation increases increases rapidly. rapidly. For For example, Rbl at at 266 266 nm nm where where 2ftw/ 2fico/Eg 1.47, the the experimentally experimentally ple, in in RbI E ==1.47, 37 is measured 2PA coefficient coefficient of of 2.49 measured 2PA 2.49 cm/bw cm/kiW 37 is 3.7 3.7 times times larger larger than than predicted by Eq. Eq. (22). (22). At At aa wavelength wavelength of of 355 355 nm nm where where 2fiw/ 2fto>/ Eg predicted by Eg== 1.10, coefficient of of 5.08 5.08 cm/GW is 14 14 37 is 1.10, however, however, the the measured measured 2PA 2PA coefficient cm/ GW 37 times predicted. This This trend trend is is maintained maintained for for the the limited limited times larger larger than than predicted. data available.37 Considering color color centers centers as as excitons available. 37 Considering excitons with large large binding binding energies, energies, such such large large correction correction factors factors may may be be accounted accounted for. for. The study. A A The role role of of excitons excitons and and color color centers centers in in 2PA 2PA needs needs further further study. study wavelength dependence study of the wavelength dependence of of 2PA 2PA near near the the gap using a 10* i 104 aa 103 103 -® •® ^ i bb f hh ©. f êu CD i dd © ® °e 10 102 - ' <D i esi N Qi OO 8g c k ——® nn © m 10 1 101 00 65 65 70 70 75 75 80 80 85 85 Year Year Fig. 17. A semilogarithmic semilogarithmic plot plot of ofthe thereported reported twotwo-photon absorption Fig. 17. A photon absorption coefficientsfor forGaAs GaAs(lower (lower-case letters) and and CdSe CdSe (upper (upper-case coefficients -case letters) -case letters) versus year: year: (a) (a) Ref. Ref. 28, 28, (b) (b) Ref. Ref. 32. 32, (c) (c) Ref. Ref. 31, (d) Ref. 38, (e) Ref. 39, (f) (g) Ref. 41, (h) (h) Ref. 27, 27. (i) Ref. 26, (k) (k) Ref. 42, (I) (I) Ref. 43, (m) (m) Ref. 41, (g) 31, (C) (C) Ref. Ref 45, 45, (D) (D) Ref. Ref. 13, 13,(E) (E) Ref. 44, (n) this work, (A) Ref. 28, (B) Ref. 31, Ref. 46, (F) (F) this work. continuously tunable tunable laser laser should should help help clarify clarify the the role role of of excitons. excitons. continuously It is is interesting interesting to to look look at at the the history history of the measurement measurement of It of the of 2PA 2PA coefficients. 17 shows coefficients. Figure Figure 17 shows experimental experimental values values of of fa ß2 reported reported for for GaAs CdSe for 1966. This GaAs and and CdSe for years years beginning beginning with with 1966. This figure figure illustrates illustrates the values as as both the lasers lasers and and the the the fairly fairly steady steady decrease decrease in in reported reported values both the experimental techniques were were refined. of the the earlier earlier data data were were experimental techniques refined. Much Much of obtained with nanosecond nanosecond pulsed pulsed lasers lasers where where photogenerated car obtained with photogenerated carrier absorption is expected expected to dominate. dominate. Additionally, Additionally, in some some rier absorption instances instances multimode multimode lasers lasers were were used. used. An An additional experimental problem previously recognized overestimate problem not not previously recognized that that can can lead lead to to an an overestimate of fa is the extreme extreme defocusing defocusing present present may may allow allow some some of of the the of 02 is that that the transmitted light to to go go undetected. undetected. transmitted light While While this this defocusing defocusing may may be be an an experimental experimental difficulty difficulty in in deter determining nonlinear optical optical devices devices mining fa, ß2, itit can can be be extremely extremely useful useful for for nonlinear such such as as the the limiter limiter discussed discussed in in Sec. Sec. 6.6. We Wefound found that that this this defocusing defocusing was quantitatively explained the nonlinear nonlinear was quantitatively explained by by attributing attributing all all of the refraction buildup of of excited excited carriers, carriers, although the the simple simple refraction to to the buildup Drude theory theory had had to to be be modified modified to transitions. Drude to allow allow for for interband interband transitions. These may actually These fits fits indicate indicatethat that interband interband transitions transitions may actually dominate dominate the refractive index index change change in in these In addiaddi the refractive these two-photon two -photon absorbers. absorbers. In tion, we demonstrated simple but but effective effective optical tion, we demonstrated aa simple optical limiter limiter based based on on two-photon the associated associated selfself-defocusing, two -photon absorption, absorption, the defocusing, and and spatial spatial filtering. By choosing parame filtering. By choosing aa substance substance with with the the proper proper material material parameters and aa specific specific geometry, geometry, this to suit suit aa ters and this limiting limiting can can be be tailored tailored to specific specific need. need. Clearly, Clearly,further further applications applications of of the the combined combined action action of of 2PA and selfself-refraction devices have 2PA and refraction in in nonlinear nonlinear optical optical devices have been been and and will will be be found. found. . 8. ACKNOWLEDGMENTS 8. ACKNOWLEDGMENTS The authors authors are are most most grateful grateful to to B. B. S. The S. Wherrett Wherrett for for many many stimulating stimulating and lengthy discussions. discussions. This and lengthy This research research was was supported supported with with funds funds from Science Foundation (ECS #8310625), #8310625), the from the the National National Science Foundation (ECS the Office Office of of Naval Research, the the Defense Defense Advanced Advanced Research Research Projects Projects Agency, Agency, Naval Research, and Robert A. A. Welch Welch Foundation. Foundation. and The The Robert 9. REFERENCES 9. REFERENCES 1. 1. E. W. W. Van E. H. Vanherzeele, Vanherzeele, M. M. J. J. Soileau, Soileau, in in Van Stryland, Stryland, H. M. A. A. Woodall, Woodall, and and M. Induced Damage Damage in Optical Optical Materials," Natl. (U.S.) "Laser Induced Natl. Bur. Stand. (U.S.) Spec. Spec. Publ., Publ., to to be be published published (1985). (1985). 2. 2. E. Kane, J. J. Chem. Chem. Phys. Phys. Solids Solids 1,1, 249 249 (1957). (1957). E. O. O. Kane, 3. C. C.R. R.Pidgeon, Pidgeon,B. B. S. S. Wherrett, Wherrett, A. A. M. M. 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