Nonlinear Optical Wave Equation for Micro- and Nano- Structured Media and Its Application

Transcription

1 AFRL-AFOR-UK-TR-13-1 Nonlinear Otical Wave quation for Micro- and Nano- tructured Media and Its Alication Dr. Valeri I. Kovalev Valeri Kovalev P. N. Lebedev Physical Institute of the Russian Academy of ciences 61 ighthill Terrace dinburgh, United Kingdom H11 4Q OARD Grant Reort Date: March 13 Final Reort for 1 October 11 to 3 etember 1 Distribution tatement A: Aroved for ublic release distribution is unlimited. Air Force Research Laboratory Air Force Office of cientific Research uroean Office of Aerosace Research and Develoment Unit 4515 Box 14, APO A 941

2 RPORT DOCUMNTATION PAG Form Aroved OMB No Public reorting burden for this collection of information is estimated to average 1 hour er resonse, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and comleting and reviewing the collection of information. end comments regarding this burden estimate or any other asect of this collection of information, including suggestions for reducing the burden, to Deartment of Defense, Washington Headquarters ervices, Directorate for Information Oerations and Reorts (74-188), 115 Jefferson Davis Highway, uite 14, Arlington, VA -43. Resondents should be aware that notwithstanding any other rovision of law, no erson shall be subject to any enalty for failing to comly with a collection of information if it does not dislay a currently valid OMB control number. PLA DO NOT RTURN YOUR FORM TO TH ABOV ADDR. 1. RPORT DAT (DD-MM-YYYY) 3. DAT COVRD (From To) March TITL AND UBTITL. RPORT TYP Final Reort Nonlinear Otical Wave quation for Micro- and Nano- tructured Media and Its Alication 1 October 11 3 etember 1 5a. CONTRACT NUMBR FA b. GRANT NUMBR Grant c. PROGRAM LMNT NUMBR 6. AUTHOR() Dr. Valeri I. Kovalev 611F 5d. PROJCT NUMBR 5d. TAK NUMBR 5e. WORK UNIT NUMBR 7. PRFORMING ORGANIZATION NAM() AND ADDR() Valeri Kovalev P. N. Lebedev Physical Institute of the Russian Academy of ciences 61 ighthill Terrace dinburgh, United Kingdom H11 4Q 9. PONORING/MONITORING AGNCY NAM() AND ADDR() OARD Unit 4515 BOX 14 APO A PRFORMING ORGANIZATION RPORT NUMBR N/A 1. PONOR/MONITOR ACRONYM() AFRL/AFOR/IO (OARD) 11. PONOR/MONITOR RPORT NUMBR() AFRL-AFOR-UK-TR DITRIBUTION/AVAILABILITY TATMNT Distribution A: Aroved for ublic release; distribution is unlimited. 13. UPPLMNTARY NOT 14. ABTRACT A novel otical wave equation is develoed for otical wave roagation and interaction in materials with intrinsic or induced macroscoic satial inhomegeneity. It shows that the effect of satial inhomogeneity is most ronounced for sizes of inhomogeneities on a sub-wavelength scale, that is in materials commonly referred to as nanohotonic materials. Resectively the new equation rovides a rigorous theoretical temlate for emerging exerimental activity worldwide in the area of linear and nonlinear nanohotonics. 15. UBJCT TRM OARD, Nonlienan Otical Wave quation, Nano-structed Media, Nonlinear Fiber Lasers 16. CURITY CLAIFICATION OF: 17. LIMITATION OF a. RPORT UNCLA b. ABTRACT UNCLA c. THI PAG UNCLA ABTRACT AR 18, NUMBR OF PAG 1 19a. NAM OF RPONIBL PRON John Gonglewski 19b. TLPHON NUMBR (Include area code) +44 () tandard Form 98 (Rev. 8/98) Prescribed by ANI td. Z39-18

3 Nonlinear Otical Wave quation for Micro- and Nano-tructured Media and Its Alication Grant No: FA Period of erformance: 1 October 11 3 etember 1 PI: Dr. Valeri I. Kovalev, Dc, Chief cientist P. N. Lebedev Physical Institute of the Russian Academy of ciences, Moscow, Russia Also with the Heriot-Watt University, dinburgh, UK. kovalev@sci.lebedev.ru; v.kovalev@hw.ac.uk Distribution A: Aroved for ublic release; distribution is unlimited. 1

4 Table of contents Page No ummary 3 1. Introduction 4. Historical background 4 a). instein s aroach 4 b). Landau s aroach 6 3. OW for satially inhomogeneous media 7 3a). OW for scattering from a scalar tye satial inhomogeneities 7 3b). OW for backscattering from a eriodic satial inhomogeneity 8 3c). Generalisation of the new OW 9 4. Conclusions 9 References 9 List of ymbols, Abbreviations, and Acronyms 1 Distribution A: Aroved for ublic release; distribution is unlimited.

5 ummary Major trends recently occurred in the modern hysics and alications of nonlinear otics are closely linked with the recent raid technological advances in material science. These are engineering of material structures of nanometre scale length and novel concets for tailoring their nonlinear resonse including substantial enhancement and tuning their nonlinearities. R&D in this subject area called nonlinear nanohotonics. Areas of alication for this are enormous. ubstantial rogress is exected to haen and is also haening in more traditional areas of R&D in nonlinear otics, such as highly efficient nonlinear frequency conversion (harmonics generation, u- and downconversion, suercontinuum generation, etc.) in smaller size devices such as hotonics crystal structures, carbon nanotubes and grahene. vidently all such alications require an adequate theoretical suort. The essential feature of the described above structures is their satial scale length which is smaller than radiation wavelength. This roerty is however not described by the Bloembergen s iconic nonlinear otical wave equation (NOW) which is only valid for macroscoically homogeneous media. In this work a theoretical formulism is develoed for otical wave linear and nonlinear roagation and interaction in materials with intrinsic or induced macroscoic satial inhomegeneity. This formulism has emerged from continuation and further develoment of the theory of light scattering by instein and by Landau and Lifshitz. The obtained otical wave equation (OW) is alicable to variety linear and nonlinear otical interactions in satially inhomogeneous media. While it is valid regardless of the satial scale of these inhomogeneities, it exlicitly shows that the strongest contribution is exected from inhomogeneities of a sub-wavelength scale size, tyical for nanostructured otical media. Therefore the obtained equation is laying a more rigorous theoretical basis for the raidly growing activity worldwide in studies of nanohotonic systems. Distribution A: Aroved for ublic release; distribution is unlimited. 3

6 1. Introduction. Phenomena of light roagation and interaction in an otically transarent nonmagnetic continuous medium is the subject area of the macroscoic electrodynamics [1,]. For the majority of these henomena the medium can be reasonably well aroximated as the satially homogeneous one. Resectively their theoretical descrition is based on OW obtained in the aroximation of macroscoically homogeneous medium [,3]. On the other hand there are lenty of henomena in otics which are inevitably associated with medium s inhomogeneites. Historically first one, Rayleigh scattering, was investigated more than a century ago by Tyndall (1869) and Lord Rayleigh (1899), who the attributed henomenon to sub-wavelength size articles (macroscoic inhomogeneities of the refractive index) in a visually transarent medium. Later, through the XX century, a variety of linear and nonlinear otical henomena were discovered and studied, in which macroscoic satial inhomogeneity of the medium is the integral art of hysics behind the henomenon. Among these are a range of sontaneous and stimulated scattering henomena, static and dynamic holograhy (including Bragg gratings), four wave mixing, etc. Most recently, mainly in the XXI century, the attention turned to otical henomena in microand nanostructured media (hotonic crystals, randomly and regularly distributed in sace nanoscale size objects (articles, rods, rings, etc.). Our analysis of ublications on studies of these henomena have revealed that their theoretical descrition is using the OW for macroscoically homogeneous media, [4], or based on various heuristic aroaches. In this work an attemt is made to rigorously consider in OW the effect of the medium s satial inhomogeneity.. Historical background a). instein s aroach Probably the first attemt to obtain OW for macroscoically inhomogeneous media was made by instein [5]. instein s rocedure of obtaining the equation is as follows. If the incident light is a quasimonochromatic electro-magnetic (M) wave with the central frequency, an otically transarent nonmagnetic ( = 1) medium can then be characterised by the corresondent to that frequency ermittivity. The ermittivity of an otically isotroic inhomogeneous medium is suosed to be the sum of a background homogeneous art,, and of a satially modulated art r), ( r ) ( r ). (1) Macroscoic electric,, and magnetic, H, fields of otical radiation at each oint of the medium are governed by the macroscoic Maxwell equations, 1 D H, () c t 1 H, (3) c t D, (4) H. (5) where c is the velocity of light and D is the electric induction which is couled with the electric field,, through the relation D. (6) Using the standard rocedure of exclusion of the magnetic field, H, one can convert qs () and (3) to the wave equation for the electric field, Distribution A: Aroved for ublic release; distribution is unlimited. 4

7 1 c D, (7) t which, along with qs(4) and (6), describe roagation of an M wave in a dielectric medium. In an otically inhomogeneous medium the electric field of the otical wave,, which is roagating in the direction of the incident to that medium otical wave, differs from the electric field of the incident wave,, by a field of the scattered wave,, so, the total electric field in the medium is. (8) When (r) and are infinitesimal comared to and, q.(7) can be slit to two couled equations for and for, the latter of which is with D 1 c D t, (9), (1) and the equation for with is 1 D c t, (11) D. (1) mall terms of the second order were neglected in obtaining qs (9)-(1). This in articular means that roagation of the um field,, is governed by the otical wave equation, q.(11), with a constant background ermittivity,. ubstituting q.(1) into qs (4) and (9) and taking into account that F ( F) F converts the equations for the scattered field to the form, c t D ( ) c t, (13) ( ). (14) Using q.(14) and taking into account that instein obtained the exression for, ( F ) F F, [6], and 1, (15) substitution of which into q.(13) has given his working OW for the scattered field, Distribution A: Aroved for ublic release; distribution is unlimited. 5

8 c t 1 ( ) c t, (16) in which the first term on the RH is roortional to, that is it exlicitly accounts the effect of satial variation of, while the second term is roortional to the magnitude of showing that it is indeendent of the rate of its satial variation. Analysing this equation instein has however noted that the first term on the RH of q.(16) is actually generating a longitudinal electric field, which cannot contribute to the scattered field, roagation of which is described by the LH of this equation. Moreover he has found that its contribution is comensated by the longitudinal comonent from the second term, [5]. Consequently the wave equation for the scattered field acquires the form, c t c t, (17) where the symbol on the RH bears witness to the fact that only transverse comonents of the exression in the brackets are contributing to the field on the LH. vidently the RH of q.(17) does not have terms accounting the effect of any kind of satial variation of. As such in its very essence it describes the scattered signal as a secondary emission of the deviations r) regardless of the rate of satial variation of these deviations, that is the scattering in satially homogeneous media too. Obviously this is in direct contradiction with the basic hysics of the Rayleigh scattering henomenon introduced/considered by Tyndall (1869) and Lord Rayleigh (1899) which exlicitly demonstrate deendence of scattering characteristics on the size of inhomogeneities. The form q.(17) is fully consistent with the iconic NOW, which was obtained by Bloembergen with colleagues for the case of otically homogeneous media [3]. The only difference is that in NOW is deendent of the otical field(s). Obviously an alternative aroach is required to obtain OW for describing the effect(s) in otically inhomogeneous media. ince we are interested in accounting for the satial effect, it can be done through the term in the wave equation for the scattered field, q.(9), with satial derivatives only. b). Landau s aroach The way to do this is actually described in [1]. Instead of relacing instein, Landau and Lifshitz (L&L) relaced under ( ) by D in q.(9), as it was done by D, (18) which follows from q.(1). Then in the aroximation of monochromatic incident and scattered waves they got the equation for the induction of the scattered radiation, D, D [ ] D. (19) c quation (19) was solved and analysed in [1] to elucidate various roerties of the intensity, I =, of scattered radiation in various media. Interestingly but in site of essential difference between the RHs of qs (17) and (19), the calculated by both instein in [5] and L&L in [1] the integral extinction coefficient due to Rayleigh scattering coincide. Probably because of this coincidence the difference between the two aroaches was not considered so far. Distribution A: Aroved for ublic release; distribution is unlimited. 6

9 3. OW for satially inhomogeneous media To understand, clarify and areciate the actual value of L&L aroach and its difference from instein s aroach we firstly must consider the cases in similar aroximations. We consider the interaction of quasi-monochromatic quasi-lane incident and scattered waves as a more general aroximation comared to the case of interaction of monochromatic lane waves considered in [1]. In this aroximation by substitution of into the first term on the LH of q.(9) we shall get the equation for the induction of the scattered radiation, D, in the form D D [ ]. () c t ince the electric field is scattered radiation is must be transverse and it is conventionally detected at large enough distance from the scattering region, D is related to there as D =, and q.() transforms to 1 [ ]. (1) c t Here again on the RH we should consider the contributions from its transverse comonents only. 3a). OW for scattering from a scalar tye satial inhomogeneities To comare qs (17) and (1) consider scattering of a quasi-monochromatic lane wave, i( tk z) ( r, ( z, e, (here ( z, is the slowly varying in sace and time amlitude of that wave) as an incident field in a medium where r) is an indeendent of time scalar function of satial coordinates. In such case, by using the roerties of the del oerator,, ( F) ( F) F, [ F G] ( G) F ( F ) G F( G) G( F) and ( ) (see ection 5.5 in [6]), we shall get [ ] ( ) ( ) ( ) ( ). () By taking into account that to q.(11) we have is a transverse electric field of the incident radiation, for which according, (3) c t q.() can be rewritten as [ ] c t ( ( ) ) ( ( ) ( ) ) ( ) ubstituting q.(4) into q.(1) we obtain the equation for the scattered field, c t c t [ 1 ( ) (. ) ( ) ]. (4) (5) Distribution A: Aroved for ublic release; distribution is unlimited. 7

10 It s easy to see that the first term on the RH of q.(5) coincides with the RH of instein s equation, q.(17). The remaining chain of four terms on the RH of q.(5) is new to the theory of wave roagation in otical media, and these are reresenting a secific contribution of a medium s macroscoic satial inhomogeneity. 3b). OW for backscattering from a eriodic satial inhomogeneity Consider then the case when the incident radiation is a lane wave of the frequency with the wavevector k, which is roagating along +z, ( r, i( tk z) e and r) is a eriodic grating with the wavevector q also along +z,, (6) iqz ( z ) ( z) e. (7) In such case the scattered radiation is also a lane wave of the frequency = with the wavevector k = k + q, the last two terms on the RH of q.(6) are zeros and the equation transforms to z k [ k q qk ]. (8) This to be comared with the result of substitution qs (6) and (7) into q.(17). Obviously the first term on the RH of this equation is zero, and therefore equation for the scattered field reduces to z k k. (9) The difference is clear: the RH of q.(8) has two terms which exlicitly deend on the rate of satial variation of, that is on q. It reduces to instein s equation for the scattered field when the medium is macroscoically homogeneous that is when the characteristic size of inhomogeneites a q -1 (q ) or when (r) is a slowly enough varying on r function, a >> ( = n/k is the wavelength of the incident radiation and n is the refractive index). When q or k the contribution of the last two terms on the RH of q.(8) can be not small. In articular when q = k (this case is a tyical for the Rayleigh and Brillouin backscattering and for the Bragg grating backreflection) the contribution from the last two terms is 6 times bigger than form the first one. Clearly the difference is much more ronounced when a <<. That case is tyical for nanostructured media, where a are conventionally in the range from ~1 nm to ~1 nm. This circumstance in articular would allow account for a substantial inconsistency between exerimental observations of some henomena in nanostructured materials and their theoretical descrition in frames of the existing theory [7]. In such media the second terms on the RH of q.(8) and of q.(5) are dominating over all other. An imortant result of this is essential simlification of the equations, and z k 1 c t q, (3) (31) which is obviously useful for ractice. Distribution A: Aroved for ublic release; distribution is unlimited. 8

11 3c). Generalisation of the new OW Obviously in a medium can vary in both sace and time, ( r,. Then scattering of an incident M wave in an inhomogeneous medium is the aearance in it of M waves, not only directions but also frequencies of which not coincide with these of the original incident wave [1]. Then numerous, mostly nonlinear, otical henomena, which conventionally were not considered being scattering, can be understood and treated as scattering. Harmonics generation, self- and cross-hase modulation, various arametric rocesses, four-wave mixing, etc. are among such henomena. In the most general case in a medium is a tensor, ˆ ( r,. In this case [ ] in qs (1) and (5) has to be a vector whose comonents are [ ik (r, k ], where ik (r, and k are the comonents of the medium s ermittivity tensor and of the um field [1]. A variation of ermittivity can be imrinted (linear scattering) or induced by the incident otical radiation (nonlinear scattering) in the medium, i.e. ˆ ( r, ˆ ( r, ˆ ( r,, (3) L where ˆ L ( r, and ˆ NL ( r, are the linear and nonlinear arts of the medium s ermittivity. When NL (r, is negligible q.(5) is describing linear light scattering and Bragg reflection without and/or with (when L (r, is deendant of time) change of the frequency sectrum (broadening and/or shif. When NL (r, is not small q.(5) describes a range of nonlinear otical henomena when the medium is induced to be macroscoically inhomogeneous (for examle stimulated scattering henomena). Thus q.(5) with q.(3) is the generalization of the iconic NOW obtained in [3], for light-matter interactions in macroscoically inhomogeneous media. 4. Conclusions A novel otical wave equation is develoed for otical wave roagation and interaction in materials with intrinsic or induced macroscoic satial inhomegeneity. It shows that the effect of satial inhomogeneity is most ronounced for sizes of inhomogeneities on a sub-wavelength scale, that is in materials commonly referred to as nanohotonic materials. Resectively the new equation rovides a rigorous theoretical temlate for emerging exerimental activity worldwide in the area of linear and nonlinear nanohotonics. References. [1]. L. D. Landau and. M. Lifshitz, lectrodynamics of Continuous Media, Addison Wesley, Reading, 196. []. J. D. Jackson, Classical lectrodynamics. John Wiley & ons, Inc., New York, 196. [3]. J. A. Armstrong, N. Bloembergen, J. Ducuing and P.. Pershan, Interaction between light waves in a nonlinear dielectric Phys. Rev. 17, (196). [4]. V. I. Kovalev, R. G. Harrison. A new nonlinear-wave-equation formalism for stimulated Brillouin scattering. Phys. Lett. A, 374, 97-3 (1). [5]. A. instein, Theorie der Oaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritischen Zustandes, Ann. Phys. 33, (191). [6]. G. A. Korn, T. M. Korn, Mathematical Handbook for cientists and ngineers, McGraw-Hill, New York [7]. A. Weber-Bargioni, rivate communication at the Conference ICN+T 1, (Paris, France). NL Distribution A: Aroved for ublic release; distribution is unlimited. 9

12 List of ymbols, Abbreviations, and Acronyms r and t the sace(osition) and time variables - the frequency of otical radiation - the medium s ermittivity - the medium s ermeability - the background homogeneous art of the medium s ermittivity r) - satially modulated art of the medium s ermittivity - the macroscoic electric field of an otical radiation wave H - the macroscoic magnetic field of an otical radiation wave D - the electric induction - the del oerator - the field of the otical wave roagating in the direction of the incident radiation - the field of the scattered wave D - the electric induction of the wave roagating in the direction of the incident radiation D - the electric induction of the scattered wave F - a vector function of the osition - a scalar function of the osition - the symbol denoting the transverse comonents I - the intensity of scattered radiation ( z, - the slowly varying in sace and time amlitude of - the frequency of the field k - the wavevector of the field +z - the direction of the incident radiation roagation (z) - the slowly varying along z amlitude of z) q - the wavevector of the grating of z) - the frequency of the scattered radiation k - the wavevector of the scattered radiation a - the characteristic size of inhomogeneites - the wavelength of the incident radiation n - the refractive index ˆ ( r, - the tensor of medium s ermittivity. ˆ ( r, and ˆ ( r, - the linear and nonlinear arts of the medium s ermittivity tensor L NL ik (r, and k - the comonents of the medium s ermittivity tensor and of the incident field OW - otical wave equation R&D - research and develoment NOW - nonlinear otical wave equation M wave - electro-magnetic wave RH - right hand side LH - left hand side L&L - Landau and Lifshitz Distribution A: Aroved for ublic release; distribution is unlimited. 1