Skip to main content
SpringerLink
Log in

Theory of nonlinear light scattering from phonons in crystals

Теория нелинейного рассеяния света на фононах в кристаллах

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

Nonlinear light scattering from phonons opens up new possibilities in the study of ionic motion. A theory of nonlinear inelastic light scattering in solids involving two incident photons of frequencies ω(in1) and ω(in2), a phonon of frequency ω(in0), and the scattered photon of frequencyω 3=ω 1+ω 2±ω 0 is developed. When both the incident light waves are the same, it reduces to the intensity-dependent light scattering from phonons in crystals. It is shown that the cross-section for the nonlinear scattering is smaller by a factor |E(inL)|(su2)/{|E(ina)|}(su2) as compared to the corresponding linear scattering with only one incident photon, where |E L| is the amplitude of the electric field of the additional incident wave andE ais of the order of (106÷107) e.s.u. at optical frequencies. A nonlinear polarizability theory for the scattering based on the linear polarizability theory of Born and Bradburn of the first-order linear Raman effect is also considered and shown to be equivalent to the general theory in the limit of zero phonon frequency. Symmetries of the scattering tensor and selection rules for long-wavelength nonpolar optical phonons are derived for all the cubic point groups. These selection rules are different from those for the first-order Raman effect and the intra-red absorption. The rate of photon production in the stimulated nonlinear process is discussed.

Riassunto

La dispersione non lineare della luce da parte dei fononi apre nuove prospettive nello studio del moto ionico. Si sviluppa una teoria della dispersione anelastica non lineare della luce nei solidi che coinvclge due fotoni incidenti di frequenzaω 1 eω 2, un fonone di frequenzaω 0, ed un fotone diffuso di frequenzaω 3=ω 1+ω 2±ω 0. Se entrambe le onde di luce incidenti sono uguali, il modello si riduce a quello della dispersione, dipendente dall'intensità, della luce da parte dei fononi in un cristallo. Si dimostra che la sezione d'urto per la dispersione non lineare è minore per un fattore |E L|2/|E a|2 di quella della corrispondente dispersione lineare con un solo fotone incidente. |E L| è l'ampiezza del campo elettrico dell'onda incidente aggiunta edE aè dell'ordine di (106÷107) e.s.u. per frequenze ottiche. Si esamina anche una teoria non lineare della polarizzabilità per la dispersione basata sulla teoria lineare della polarizzabilità di Born e Bradburn per l'effetto Raman lineare del primo ordine, e si dimostra la sua equivalenza con la teoria generale nel limite nullo della frequenza del fonone. Si ricavano per tutti i gruppi di punti cubici le simmetrie del tensore di dispersione e le regole di selezione per fononi ottici non polari di grande lunghezza d'onda. Le regole di selezione ottenute sono differenti da quelle dell'effetto Raman del primo ordine e dell'assorbimento nell'infrarosso. Si discute la produzione dei fotoni nei processi non lineari stimolati.

Резюме

Неупругое рассеяние света фононами открывает новые возможности для исследования движения ионов. Развивается теория нелинейного неупругого рассеяния света в твердых телах, включающая два падающих фотона с частотамиω 1 иω 2, фонон с частотойω 0 и рассеянный фотон с частотойω 3=ω 1+ω 2±ω 0. Когда обе падающие световые волны являются одинаковыми, этот процесс сводится к рассеянию света на фононоах в кристаллах, которое зависит от интенсивности. Показывается, что поперечное сечение для нелинейного рассеяние только с одним падающим фотоном где |E l| представляет амплитуду электрического поля дополнительной падающей волны иE aимеет порядок (106÷107) единиц СГСЭ при оптических частотах. Также рассматривается нелинейная теория поляризуемости для рассеяния, основанная на линейной теории поляризуемости Борна и Брадбурна для линейного эффекта Рамана первого порядка. Показывается, что рассматриваемая теория эквивалентна общей теории в пределе нулевой частоты фонона. Для всех кубических точечных групп выводятся симметрии тензора рассеяния и правила отбора для длинноволновых неполлярных оптических фононов. Эти правила отбора отличаются от правил отбора в эффекте Рамана первого порядка и в инфракрасном поглощении. Обсуждается интенсивность рождения фотона в вынужденном нелинейном процессе.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. SeeR. Loudon:Adv. Phys.,13, 423 (1964).

    Article  ADS  Google Scholar 

  2. R. W. Terhune, P. D. Maker andC. M. Savage:Phys. Rev. Lett.,14, 681 (1965).

    Article  ADS  Google Scholar 

  3. S. J. Cyvin, J. E. Rauch andJ. C. Decius:Journ. Chem. Phys.,43, 4083 (1965) See alsoY.-Y. Li andWu Li Hsueh Pao:Acta Phys. Sinica 20, 164 (1964);S. Kielich:Bull. Acad. Polon. Sci. Ser. Math., Astr., Phys.,12, 53 (1964).

    Article  ADS  Google Scholar 

  4. Wherever it is meaningful, we will use the notation in which—ω corresponds to destruction and+ω corresponds to creation.

  5. One must be careful in using this interpretation. In the electric-dipole approximation, which we will use throughout this paper, for solids with inversion symmetry,\(\mathop {\chi _{lmn} }\limits^{ NL} = 0\). This implies that there is no nonlinear elastic scatteringω 3=ω 1+ω 2±ω 0 in such solids. However, because of phonons, the nonlinear (inelastic) scattering considered in this paper is not zero. (See Sect.4).

  6. SeeN. Bloembergen:Nonlinear Optics (New York, 1965).

  7. M. Born andM. Bradburn:Proc. Roy. Soc., A188, 161 (1947).

    Article  ADS  Google Scholar 

  8. R. Loudon:Proc. Roy. Soc., A275, 218 (1963).

    Article  ADS  Google Scholar 

  9. See,e. g.,J. M. Ziman:Electrons and Phonons (London, 1960);H. Ehrenreich:Journ. Phys. Chem. Solids,2, 131 (1957).

  10. G. D. Whitfield: Phys. Rev.,121, 720 (1961). See also ref. (7).M. Born andM. Bradburn:Proc. Roy. Soc., A188, 161 (1947).

    Article  ADS  MathSciNet  Google Scholar 

  11. In the absence of spin-orbit interaction, the Bloch wave functions can always be chosen so that\(\Psi _{b^k } = \Psi _{b - k}^ * \). This impliesp j*bb′ (k)=−p j*b′b (−k)=−p jbb′ (−k), andG i* b′b (k)=G i* bb′ (−k)=G i b′b (−k)for optical modes. SinceE b k=E b-k,from eq. (2.21) one obtainsS i*123 =S *123 for optical phonons. For the acoustic modeG i* b′b (k)=−G i b′b (−k) in our notation.

  12. See,e. g. ref. (7).

    Article  ADS  Google Scholar 

  13. The first-order Raman tensor defined here is real, dimensionless and of the order of unity for the nonresonant scattering. It is related to the tensorR i12 defined by Loudon (ref. (1.7)) byR i12 =nmℏ2 R i12 , which in turn is related to the derivative of the linear polarizability ∝12,i (see ref. (1)) byR i12 (−ω1,ω1,0)=−(ω1ω1 m 2ℏ2 a/e 2)∝12,i (ω1).

  14. See,e. g.,P. N. Butcher andT. P. McLean:Proc. Phys. Soc.,81, 219 (1963);J. A. Armstrong, N. Bloembergen, J. Ducuing andP. S. Pershan:Phys. Rev.,127, 1918 (1962);S. S. Jha andC. S. Warke:Nuovo Cimento,53 B, 120, (1968).

    Article  ADS  MathSciNet  Google Scholar 

  15. The derivation of phonon selection rules closely follows the method used byOvander (L. N. Ovander:Optics and Spectroscopy,9, 302 (1960) for the first-order linear Raman scattering. See alsoO. Theimer:Canad. Journ. Phys.,34, 312 (1956);S. Bhagavantam andT. Venkatarayudu:Proc. Indian Acad. Sci., A9, 224 (1939).

    ADS  Google Scholar 

  16. See,e.g.,L. D. Landau andE. M. Lifshitz:Quantum Mechanics (London, 1958), p. 336.

  17. O. V. Kovalev:Irreducible Representations of the Space Groups (New York, 1965). We use more standard names, developed by chemists, for the representations instead of that given in this book.

  18. SeeR. Loudon: inProceedings of the International Conference on Light Scattering Spectra of Solids, New York, 1968 (to be published).

  19. (19)See,e.g.,J. M. Worlock andP. A. Fleury:Phys. Rev. Lett.,19, 1176 (1967).

    Article  ADS  Google Scholar 

  20. Selection rules for infra-red and Raman spectroscopy are contained in,e.g.,H. Jaffe, andM. Orchin:Symmetry in Chemistry (New York, 1967). Note that the selection rules for nonlinear scattering derived in the previous Section are more general that those obtained in ref. (3) where ω1=ω2 and ω0=0 have been assumed.

  21. J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra andR. P. Bauman:Phys. Rev.,173, 851 (1968).

    Article  ADS  Google Scholar 

Download references

Author information

Author notes

  1. J. W. F. Woo

    Present address: Tata Institute of Fundamental Research, Bombay

Authors and Affiliations

  1. IBM Watson Research Center, Yorktown Heights, N.Y.

    S. S. Jha & J. W. F. Woo

Authors
  1. S. S. Jha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. W. F. Woo
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Переведено ребакцией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jha, S.S., Woo, J.W.F. Theory of nonlinear light scattering from phonons in crystals. Nuov Cim B 2, 167–183 (1971). https://doi.org/10.1007/BF02723081

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02723081

Search

Navigation