Chapter 12. Interaction of Light and Sound

1. Chapter 12. Interaction of Light and Sound 12.0 Introduction Acousto-Optic(AO) effect : # Effect of change in the index of refraction of medium (crystal) by an Acoustic wave # Acoustic wave  Photoelastic effect  Change in refractive index Reference : A. Ghatak, K. Thyagarajam, “Optical Electronics”, Cambridge Univ. Press A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons

2. Photoelastic effect : Mechanical strain  Index of refraction Index ellipsoid for Principal axes : Strain tensor elements, S : : Normal strain : Shear strain where,u, v, w : displacements along the x, y, z axes

3. Change in index of refraction due to the mechanical strain : where,Pij : Elasto-Optic (Strain Optic) Coefficient (6x6 matrix)  Table 9.1 / 9.2 The equation of the index ellipsoid in the presence of a strain field :

4. Example) Sound wave propagating along the z direction in water Sound wave : Elasto-Optic Coefficient for the water (isotropic, Table 9.1) : The new index ellipsoid :

5. Example) y-polarized Shear wave propagating along the z direction in Ge Sound wave : Elasto-Optic Coefficient for the Ge (cubic, Table 9.1) : The new index ellipsoid :

6. Acousto-Optic effect Bragg diffraction & Raman-Nath diffraction Vector Representation light wave acoustic wave # Spread angle of Acoustic wave : Raman-Nath diffraction : acoustic wave vector has an angular distribution Bragg diffraction : acoustic wave vector is well defined # Diffraction angle of Light : # Dimensionless parameter :

7. Example) Water, n=1.33, W=6MHz (vs=1,500 m/s), l=632.8 nm Bragg diffraction : Single order diffraction Raman-Nath diffraction : Multiple order diffraction

8. Raman-Nath diffraction Moving periodic refractive index grating : Consider L is small enough so that the medium behave as a thin phase grating, where,f1=(2p/l)n0L, f1=(2p/l)Dn0L The transmitted field on the plane x=L :

9. amplitude reduction Frequency : Wave vector : Propagation in x>L : : +1 order : -1 order Diffraction angles :

10. m-th order diffractive wave : # Frequency : # Diffraction angle : # : The restriction on length of medium is severe at higher frequency  Diffraction efficiency reduction : First order diffraction maximum

11. Bragg diffraction In this regime, we can no longer consider the refractive index perturbation to act as a thin phase grating. We should consider the propagation equation of light ;

12. Let, And, slow varying approximation ;

14. (1) Small Bragg angle diffraction : Bragg condition

15. These equations have a solution for when only , The solutions for a+=a, a-=a are independent each other, so let a+=a

16. Diffraction efficiency, h 0-th and 1-st diffraction powers : i) ii) Maximum transfer : Diffration efficiency :

17. Acoustic intensity : (Text p. 483) : Figure of Merit Diffraction efficiency :

18. Acoustic intensity for maximum efficiency : Diffraction figure of merit of the material relative to water Acoustic power for maximum efficiency (LH cross-section, maximum impedance matching case) :

19. (2) Large Bragg angle diffraction x-dependent term 

20. These equations have a solution for when only , The two solutions are independent each other, so let 1) b+=b+K (Co-directional coupling) Solutions :

21. Diffraction efficiency, h 0-th and 1-st diffraction powers : Diffraction efficiency :

22. 2) b-=b-K (Counter-directional coupling) Solutions :

23. # Application : DBR reflector

24. Surface Acousto-Optics : Diffraction effect through a thin film surface or wave guide : High intensity localized on the interface  enhancing the diffraction efficiency : 1967, Ippen et al. (first experimental demonstration in a quartz) Surface undulation profile by the acoustic wave : Wave vectors of reflected and transmitted light waves : Electric field on the surface :

25. 1) Reflection wave

26. Diffraction angle : the amplitude of reflected wave :

27. Diffraction efficiency of l-th order :

28. 2) Transmitted wave Similarly, Diffraction efficiency of l-th order :