Breakdown of the Slowly Varying Amplitude Approximation: Generation of Backward Traveling Second Harmonic Light

Transcription

1 Claremont Colleges Claremont All HMC Faulty Publiations and Researh HMC Faulty Sholarship Breakdown of the Slowly Varying Amplitude Approximation: Generation of Bakward Traveling Seond Harmoni Light J. Z. Sanborn '01 Harvey Mudd College C. Hellings '0 Harvey Mudd College Thomas D. Donnelly Harvey Mudd College Reommended Citation J. Z. Sanborn, C. Hellings, T. D. Donnelly, "Breakdown of the slowly varying amplitude approximation: generation of bakward traveling seond harmoni light," J. Opt. So. A. B, 0, (003). doi: /JOSAB This Artile is brought to you for free and open aess by the HMC Faulty Sholarship at Claremont. It has been aepted for inlusion in All HMC Faulty Publiations and Researh by an authorized administrator of Claremont. For more information, please ontat sholarship@u.laremont.edu.

2 15 J. Opt. So. Am. B/ Vol. 0, No. 1/ January 003 Sanborn et al. Breakdown of the slowly-varying-amplitude approximation: generation of bakward-traveling, seond-harmoni light J. Z. Sanborn, C. Hellings, and T. D. Donnelly Department of Physis, Harvey Mudd College, Claremont, California Reeived Deember 0, 001; revised manusript reeived August, 00 By numerially solving the nonlinear field equations, we simulate seond-harmoni generation by laser pulses within a nonlinear medium without making the usual slowly-varying-amplitude approximation, an approximation whih may fail when laser pulses of moderate intensity or ultrashort duration are used to drive a nonlinear proess. Under these onditions we show that a bakward-traveling, seond-harmoni wave is reated, and that the magnitude of this wave is indiative of the breakdown of the slowly-varying-amplitude approximation. Conditions neessary for experimental detetion of this wave are disussed. 003 Optial Soiety of Ameria OCIS odes: , , The slowly-varying-amplitude approximation (SVAA) is regularly used to simplify the analysis of light s interation with nonlinear optial media. 1, The SVAA allows one to assume that the amplitude of the light inident on the optial medium hanges negligibly over distane sales roughly equal to the light s wavelength or, equivalently, over time sales roughly equal to the light s period. This simplifies the analysis greatly beause it rids the theoretial treatment of all seond-order and higher time and distane derivatives when ompared with the muh larger first-order terms. This approximation is invoked nearly universally to desribe nonlinear optial phenomena suh as harmoni generation, stimulated light sattering (e.g., Raman and Brillouin sattering), fourwave mixing, self-fousing, and nonlinearities in waveguides. 1 3 It is well known that making the SVAA is equivalent to ignoring a bakward-traveling wave propagating at the fundamental frequeny. The SVAA begins to fail when the inident light is moderately intense or when, in the ase of ultrashort laser pulses, the pulse envelope hanges signifiantly over the ourse of a single optial yle. The inability to use the SVAA signifiantly ompliates the analysis of many nonlinear interations, rendering it impossible to form an analyti solution to the nonlinear wave equation. However, many studies have investigated impliations of the SVAA breaking down, 4 9 and in the purely time-varying domain (where it is known as the rotating-wave approximation) analyti orretions to it are known. 10,11 Casperson 6 8 has shown that instabilities are predited in steady-state laser amplifiers if the SVAA is not used. In a study of how femtoseond pulses interat with a twolevel atom Hughes 9 finds the prodution of highfrequeny spetral omponents on the propagating pulse whih are not predited when the SVAA is made. Brabe and Krausz 5 show that, in pratie, many nonlinear phenomena an be aurately desribed using the SVAA even if the driving pulse is a single yle of radiation. However their analysis does not apply to the bakwardtraveling radiation disussed here. Here we study numeri solutions to the nonlinear wave equation without making the SVAA. We fous on a partiular seond-order effet, the generation of a bakwardtraveling, seond-harmoni wave, to quantify the physial importane of the seond-order terms ignored by the SVAA. Analytial solutions that demonstrate the existene of the bakward-traveling, seond-harmoni wave have been found but require a set of onditions that are hard to realize experimentally. Speifially the urrent studies of this phenomenon assume phase mathing in both forward and reverse diretions, as well as that no attenuation of the pump field ours within the optial medium. 11 Although these assumptions may indeed be appliable to speial lasses of media, they are invalid in the general ase. In our numerial approah to this problem the above assumptions beome unneessary. Therefore the following analysis develops a more feasible method of experimentally and quantitatively studying the bakward-traveling, seond-harmoni waves predited by the numerial approah. In the following analysis we find steady-state solutions to the nonlinear wave equation that assume an inident plane wave. The results we obtain are for steady-state onditions and thus will apply preisely to ontinuous wave lasers and hold approximately for laser pulses of adequately long duration l/v, where is the pulse width, v is the group veloity of the fundamental pulse, and l is the harateristi interation length of the medium. The field equations desribing seond-harmoni generation (SHG) in steady state an be derived from Maxwell s equations when a nonlinear polarizability P NL of the medium is assumed to be present. 3 We start with Ẽ n Ẽ t 4 P NL t. (1) /003/ $ Optial Soiety of Ameria

3 Sanborn et al. Vol. 0, No. 1/January 003/J. Opt. So. Am. B 153 We assume a steady-state solution for the field, varying in time as Ẽ 1 z,t E 1 z exp i 1 t.., () and we fator out the rapidly varying part of the field aording to E 1 z A 1 z exp ik 1 z. (3) The seond-order polarizations are given by P 1 z 4d eff E 1 * z E z, (4) P z d eff E 1 z. (5) These polarizations provide the field with a soure of new omponents, partiularly the seond-harmoni wave that is the fous of this disussion. Using the above definitions, we an express the wave Eq. (1) as the two, oupled, nonlinear differential equations d A 1 da 1 dz ik 1 dz d A da dz ik dz 16d eff 1 A 1 *A exp i kz, 8d eff A 1 exp i kz, where k is the phase mismath (k 1 k ) between the forward-traveling fundamental and seond-harmoni waves and the indies i 1, orrespond to the fundamental and seond-harmoni waves. We introdue a unitless distane parameter z/l for whih the harateristi interation length l is given by 3 l d eff n 1 n 3 I (6) (7), (8) where n i is the index of refration for the ith harmoni frequeny omponent, 1 is the field s fundamental frequeny, d eff is the medium s effetive seond-order suseptibility, and I is total intensity (we know from the Manley Rowe relations that I I 1 I is spatially invariant in seond-harmoni generation). In a linear medium the omplex amplitude of a plane wave is onstant. This is not the ase in a nonlinear medium, thus it is useful to represent the omplex, slowlyvarying wave amplitude as A i ( I/n i ) u i exp(i i ). Using this substitution in Eqs. (6) and (7) and separating real and imaginary parts, we obtain a 1 u 1 a 1 u 1 1 u 1 1 a 1 u 1 1 a 1 u 1 1 u 1 a u a u u a u a u u u 1 u os s 1, (9) u 1 u sin s 1, (10) u 1 os s 1, (11) u 1 sin s 1, (1) where a i k i l and the primes indiate derivatives with respet to. These four equations are solved numerially using a Runge Kutta 4,5 algorithm assuming phase mathing only in the forward diretion s kl 0. It an be seen from Eq. (8) that the harateristi interation length l depends both on the intensity of the laser pulse and the effetive seond-order suseptibility d eff. As either the intensity or d eff is inreased the interation length dereases, thereby dereasing the distane in whih the fundamental and seond-harmoni fields exhange energy. Consequently if the interation length gets to be roughly the same size as the wavelength of the field, the SVAA A/ z k (A/ z) beomes less valid and the seond-order terms usually negleted through the SVAA may beome important. An intense titanium:sapphire pulse ( Hz, I W/m ) traveling through a lithium niobate rystal (d eff esu, n 1 n ) will have an interation length l 6 m; if the nonlinear rystal is KDP (d eff esu, n 1 n 1.5) the interation length is longer, l 00 m. Therefore for the lithium niobate rystal the steady-state approximation ( l/v) will hold for 40 fs a reasonable assumption for most laser pulses apable of reahing the required intensity. Notie that the parameters a i have the effet of ontrolling the relative strength of the seond-order terms. Thus making the SVAA is equivalent to setting a 1 and a equal to zero in Eqs. (9) (1), and in this ase the wellknown solutions for SHG are reovered and no bakwardtraveling wave is predited to exist. To obtain the magneti field from the numerial solutions of Eqs. (9) (1) we use Faraday s law to obtain a relation between Ẽ and B as whih redues to Ẽ z, t z Ẽ z, t z 1 B z, t, (13) t i 1 H z, t, (14) where B 1 H. H an be written in terms of a slowlyvarying field amplitude M(z), muh as we previously did in Eq. (3), to obtain H i z, t Eq. (14) then redues to H i z exp i i t M i z exp ik i z exp i i t. (15) aa ia im. (16) If we separate A and M into their real and imaginary parts in whih A A r ia i and M M r im i, we obtain aa r aia i ia r A i im r M i. (17)

4 154 J. Opt. So. Am. B/ Vol. 0, No. 1/ January 003 Sanborn et al. We an separate this into its real and imaginary parts to obtain M r aa i A r, (18) M i aa r A i, (19) providing us a formula for the magneti field omponents in terms of the wave amplitudes A. To eluidate the underlying physis of Eqs. (9) (1) we separate the intensities of the fundamental and seondharmoni waves into forward- and bakward-traveling omponents using methods similar to Casperson. 6 This is possible if we speify both the magneti and eletri fields. The eletri field an be deomposed into forward- ( ) and bakward- ( ) traveling omponents by noting that the eletri field an be expressed as a superposition of forward- and bakward-traveling waves in the form Ẽ z,t E z exp ikz i t E z exp ikz i t E z E z exp ikz exp ikz i t. (0) Similarly, the magneti field an be represented as H z, t 1 1 H z H z exp ikz exp ikz i t. (1) By omparing Eqs. (0) and (), we see that the field amplitudes E(z) an be redued to plus and minus omponents aording to A z E z E z exp ikz, () and similarly M z H z H z exp ikz. (3) In general the plus and minus omponents of the fields do not separately satisfy the nonlinear wave equation, only their sum does. However, we onsider a region in the nonlinear medium small enough that the forward and bakward omponents are exhanging a negligible amount of energy. The forward and bakward omponents are then well defined as independent eletromagneti fields 6 and we an use Ampere s and Faraday s laws to relate the eletri and magneti field omponents as or E z H z, (4) E z H z, (5) A z M z, (6) A z M z. (7) Using the above relations we rewrite Eq. () as A z A z A z exp ikz, (8) and Eq. (3) as M z A z A z exp ikz. (9) Combining Eqs. (8) and (9) we obtain the plus and minus omponents of the wave amplitudes for the eletri field as A z 1 A z M z, (30) A z 1 A z M z exp ikz, (31) and for the magneti field as M z 1 M z A z, (3) M z 1 M z A z exp ikz. (33) From the Poynting vetor we an now obtain the forwardand bakward-traveling intensities 8 as I 4 A r M r A i M i. (34) Now by using the plus and minus omponents of the eletri and magneti fields given by Eqs. (30) and (31), respetively, with Eq. (34) we obtain expressions for the instantaneous forward- and bakward-traveling wave intensities in the form A r M r 1 4 A i M i, (35) A r M r 1 4 A i M i. (36) Sine our numerial model will not be omputing B(z), we instead express (35) and (36) solely in terms of the wave amplitudes A(z) for the eletri field by using Eqs. (18) and (19). We then obtain two expressions for the forward- and bakward-traveling wave intensities as 4 A r aa i A i aa r, (37) 4 aa i aa r. (38) Therefore the real and imaginary omponents of A(z) are all that is required to ompute the intensity of the forward and bakward traveling waves. For instane the intensity of the bakward-traveling wave is given simply by the spatial derivatives of the eletri field at the point of interest, as expeted sine for a linear medium A i 0. As noted previously, setting a i to zero is equivalent to invoking the SVAA, and Eq. (38) shows that in this ase the bakward-traveling wave vanishes. Conversely the magnitude of the bakward-traveling wave is indiative of how severely the SVAA breaks down. Knowing the form of the intensities of the forward- and bakward-traveling waves, we now analyze the physially imposed boundary onditions that are required for the numerial analysis. To solve Eqs. (9) (1), boundary onditions must be established. We assume a plane wave inident on a non-

5 Sanborn et al. Vol. 0, No. 1/January 003/J. Opt. So. Am. B 155 linear medium of length L, as shown in Fig. 1. An antirefletive oating is assumed on the rystal s output so that no forward-traveling waves are refleted from the output fae of the rystal; therefore I 1 (L) 0 and I (L) 0. The forward intensities of the fundamental and seond harmoni are normalized to unity at the output, onstraining another boundary value I (L) 1 I 1 (L). At the input fae of the rystal z 0, we impose the physial onstraint that there be no forward-traveling seond harmoni present; that is, I (0) 0. This is beause at the input there is not yet any medium in whih to transfer energy from the fundamental field to the seond harmoni. We further require that the first derivative of the phases be set to zero at the output fae sine, from a physial standpoint, the phases must go smoothly to onstants when exiting the material; therefore 1 (L) 0, (L) 0. These six boundary onditions are the only onstraints we may impose on the system, leaving two more to fulfill the eight required for a seond-order, four-variable, boundary value problem. We an simplify our analysis by noting that Eqs. (9) (1) depend initially only on the differene of the fundamental and seond-harmoni phase, whih allows a single boundary ondition to represent their overall initial differene 1. Sine we have more onstraints at the output of the rystal than at the input, we will simplify the numerial analysis by solving Eqs. (9) (1) bakwards, from the output to the input fae. The remaining two boundary onditions at the output fae must be adjusted reursively until the onstraint that no forward-traveling, seondharmoni wave is present at the input fae is satisfied. The boundary onditions to be iteratively adjusted are the values of I 1 (L) and at the output fae. We use a numerial algorithm to find the values of I 1 (L) and at the output that satisfy the boundary ondition at the input. In the first iteration, it takes the parameter a 1 and the user s guesses for the initial values of I 1 (L) and and uses them to solve Eqs. (9) (1) bakwards from the output fae. It then finds the loation of Fig. 1. Box represents the nonlinear medium and the fundamental wave is inident from the left. As disussed in the text, the eight boundary onditions applied for the numeri analysis are (1) I 1 (L) 0, () I (L) 0, (3) I (L) 1 I 1 (L),(4) I (0) 0, (5) 1 (L) 0, (6) (L) 0, (7) 1?, and (8) I 1 (L)? A question mark denotes a boundary ondition that is unknown and must be found self-onsistently. the minimum intensity of the forward-traveling seond harmoni within its first solution, whih is an estimate for the loation of the rystal s input fae (i.e., where L 0). This length is the region that will be integrated over to find subsequent solutions. The algorithm will now reursively alter the value of to find the loal minimum value of I (0) with the given user-defined parameters I 1 (L) and a 1. It does this by stepping the value of in the diretion that dereases the value of I (0), halving the step size when near the loal minimum to find the loal minimum within a given tolerane. This tolerane is set at 0.001%, meaning that a 0.001% hange in the value of I (0) from one iteration to the next will satisfy the tolerane, and the loal minimum will have been found to the desired auray. When the algorithm determines the loation of the minimum of I (0) in phase spae, it outputs the value of I (0), the phase angle at whih the minimum ours, and the length L of the rystal. At this point the user deides if the passed-in value of I (L) must be hanged to ahieve either a lower value of I (0) or a different rystal length L (for example one that onforms with an experiment). With the boundary onditions found using the numerial algorithm as desribed above we an study the generation of the bakward-traveling, seond-harmoni wave. Figure shows the forward- and bakward-traveling waves alulated for the fundamental and seondharmoni waves when a Figure (a) shows the usual exhange of energy between the forward-traveling fundamental and seond-harmoni waves whih ours over the first few interation lengths. Figure (b) demonstrates the growth of bakward-traveling fundamental and seond-harmoni waves. The higher-order osillations result from standing waves within the nonlinear rystal. The low effiieny of bakward SHG shown in Fig. (b) is a result of the mismath in the phases of the fundamental and seond-harmoni fields in the bakward diretion. It is possible to math the phases in the forward diretion both experimentally and numerially (by setting s 0), but the total phase mismath in the bakward diretion is given by (/a 1 ) (1/a ), whih shows the impossibility of phase mathing. However, it has been shown that by using periodially poled rystals it is possible to quasi-math the phases in the bakward proess By quasi-mathing the bakward proess more growth of the bakward-traveling wave is allowed, thereby resulting in a more effiient proess ( 8% onversion effiieny by poling lithium niobate with a m period). Current analytial approximations for bakward SHG 1 depend on quasi-mathing of phases (and therefore the use of periodially-poled rystals) and negligible depletion of the pump field, two assumptions not required by the numerial analysis presented in this paper. Figure 3 shows the bakward SHG effiieny (normalized to the input fundamental wave) as a funtion of the parameter a 1. This shows an inrease in the seondharmoni effiieny as a 1 inreases. Calulations are performed only up to values of a beause materials will ertainly sustain damage at orresponding intensities above this value. For example we know from Eq. (8) and the definition of a 1 that

6 156 J. Opt. So. Am. B/ Vol. 0, No. 1/ January 003 Sanborn et al. Fig.. (a) Forward- and (b) bakward-traveling waves plotted as a funtion of for a The growth of a bakwardtraveling, seond-harmoni wave is evident in (b) as is the existene of a bakward-traveling wave at the fundamental frequeny. If the nonlinear medium is taken to be lithium niobate (d eff esu) then for typial laser parameters ( Hz, I 10 1 W/m ) the unitless length 3 orresponds to a rystal length of 18 m. Note that if the the SVAA approximation is invoked, no bakward-traveling waves exist. I n 1 4 n 3 3 d a 1, (39) eff and therefore in lithium niobate a orresponds to an intensity of approximately W/m. On the other end of the sale, for lithium niobate a orresponds to an intensity of approximately W/m. Figure 3 suggests the feasibility of experimentally verifying the breakdown of the SVAA: The growth of the bakward-traveling, seond-harmoni wave in a nonlinear medium an be monitored as a funtion of the input laser intensity and quantitatively ompared to the predited effiienies. Suh an experiment to test the breakdown of the SVAA and the numeri modeling presented here would use lithium niobate as the nonlinear medium. Lithium niobate has a large d eff ompared with KDP or BBO, meaning that to reah a nominal a value, and therefore a desired effiieny in generating a bakward-traveling, seond-harmoni wave, a lower laser intensity an be used for lithium niobate than for KDP or BBO. For example to reah an effiieny of 10 5 for the bakwardtraveling, seond-harmoni wave, a value of approximately a is neessary (see Fig. 3). This implies a laser intensity of 10 1 W/m for lithium niobate and W/m for KDP, whih is above the damage threshold. Also any possibility of self-fousing should be avoided so that knowledge of the pump laser s intensity an be ensured. This onsideration, together with onsideration of damage thresholds, further onstrains the hoie of nonlinear media. To minimize pump power (to avoid self-fousing) a material is neessary that responds relatively strongly in seond order to lower intensities, suggesting lithium niobate with a large d eff as the material of hoie. (Note that while the ritial power for selffousing depends inversely on d eff, reahing a partiular hoie of a value depends quadratially on d eff and therefore there is an advantage to hoosing materials of large d eff.) Finally a larger d eff implies a shorter interation length l and thus relatively short steady-state time. For onditions listed above ( Hz, I W/m ) lithium niobate has an interation length of 6 m and KDP has an interation length of 00 m; this implies l n 40 fs while KDP 1 ps. Using lithium niobate therefore means that our analysis would apply if typial titanium:sapphire laser pulses (pulse duration 40 fs) were used to pump the nonlinear interation. Thus a simple experiment an be performed using a titanium:sapphire pump laser and a lithium niobate rystal. The lithium niobate rystal should be antirefletion oated so that the boundary onditions listed in Fig. 1 are met. The pump laser an be foused onto a rystal whih is misaligned from normal inidene (this avoids olleting spurious bak refletions along with the desired bakward-generated, seond-harmoni light), thus generating bakward-going, seond-harmoni light whih an be olleted with a dihroi beam splitter. The signal from the bakward-traveling light an be monitored using a photomultiplier tube and ompared to the intensity of the pump laser. In this way the preditions shown in Fig. (3) an be tested. Fig. 3. Bakward SHG effiieny versus parameter a 1. At pump-laser intensities above W/m the bakward seond harmoni an be deteted. The alulation is trunated for laser intensities greater than W/m sine damage of the nonlinear medium would be expeted for higher intensities. The intensity axis is alulated assuming lithium niobate as the nonlinear medium.

7 Sanborn et al. Vol. 0, No. 1/January 003/J. Opt. So. Am. B 157 In summary, we have shown that in a moderate intensity regime the seond-order terms in the nonlinear wave equation must be inluded to model the full response of the nonlinear system. The emergene of a bakwardtraveling, seond-harmoni wave is analyzed. Its generation effiieny is alulated and related to the breakdown of the SVAA. An experiment to measure the bakward-traveling, seond-harmoni wave is outlined. ACKNOWLEDGMENTS We thank Lee Casperson and Andrew Bernoff for helpful onversations. Aknowledgment is made to the Donors of The Petroleum Researh Fund, administered by the Amerian Chemial Soiety, for partial support of this researh, and for support under an award from the Researh Corporation. T. D. Donnelly may be reahed by at tom donnelly@hm.edu. REFERENCES AND NOTES 1. J. A. Armstrong, N. Bloembergen, J. Duuing, and P. S. Pershan, Interations between light waves in a nonlinear dieletri, Phys. Rev. 17, (196).. Y. R. Shen, The Priniples of Nonlinear Optis (Wiley, New York, 1984). 3. R. W. Boyd, Nonlinear Optis (Aademi, Boston, Mass., 199). 4. S. E. Harris, Proposed bakward wave osillation in the infrared, Appl. Phys. Lett. 9, (1966). 5. T. Brabe and F. Krausz, Nonlinear optial pulse propagation in the single-yle regime, Phys. Rev. Lett. 78, (1997). 6. L. W. Casperson, Field-equation approximations and amplifiation in high gain lasers: numeri results, Phys. Rev. A 44, (1991). 7. L. W. Casperson, Field-equation approximations and amplifiation in high gain lasers: analyti results, Phys. Rev. A 44, (1991). 8. L. W. Casperson, Field-equation approximation and the dynamis of high-gain lasers, Phys. Rev. A 43, (1991). 9. S. Hughes, Breakdown of the area theorem: arrier-wave rabi flopping of femtoseond optial pulses, Phys. Rev. Lett. 81, (1998). 10. F. Bloh and A. J. Siegert, Magneti resonane for nonrotating fields, Phys. Rev. 57, 5 57 (1940). 11. For a disussion of the rotating-wave approximation in the ontext of optial phenomena see L. Allen and J. H. Eberly, Optial Resonane and Two-Level Atoms (New York, Dover, 1987), Chap.. 1. Y. J. Ding, J. U. Kang, and J. B. Khurgin, Theory of bakward seond-harmoni and third-harmoni generation using laser pulses in quasi-phase-mathed seond-order nonlinear medium, IEEE J. Quantum Eletron. 34, (1998). 13. X. Gu, R. Y. Korotkov, Y. J. Ding, J. U. Kang, and J. B. Khurgin, Bakward seond-harmoni generation in periodially poled lithium niobate, J. Opt. So. Am. B 15, (1998). 14. X. H. Gu, M. Makarov, Y. J. Ding, J. B. Khurgin, and W. P. Risk, Bakward seond-harmoni and third-harmoni generation in a periodially poled potassium titanyl phosphate waveguide, Opt. Lett. 4, (1999).