Chapter 6. Processes Resulting from the Intensity-Dependent Refractive Index

1. Chapter 6. Processes Resulting from the Intensity-Dependent Refractive Index - Optical phase conjugation - Self-focusing - Optical bistability - Two-beam coupling - Optical solitons • Photorefractive effect (Chapter 10) • : cannot be described by a nonlinear susceptibility c(n) for any value of n Reference : R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.

2. 6.1 Optical Phase Conjugation : Generation of a time-reversed wavefront Signal wave : Phase conjugate wave : where, : amplitude reflection coefficient

3. Properties of phase conjugate wave : 1) : The polarization state of circular polarized light does not change in reflection from PCM Ex) i) In reflection from metallic mirror [p-phase shift] ii) In reflection from PCM [p-phase shift & y component : ip/2  -ip/2 (-p : delayed)] 2) : The wavefront is reversed 3) : The incident wave is reflected back into its direction of incidence

4. Aberration correction by optical phase conjugation Wave equation : Solution : With slow varying approximation, Since the equation is generally valid, so is its complex conjugate, which is given explicitly by Solution : : A wave propagating in the –z direction whose complex amplitude is everywhere the complex of the forward-going wave

6. Phase Conjugation by Degenerated Four-Wave Mixing New wave (k4) source term ! 1) Qualitative understanding Four interacting waves : Nonlinear polarization : Counter-propagating waves) So,A4 is proportional to A3* (complex conjugate of A3) and its propagation direction is –k3(in the case of perfect phase matching)

7. 2) Rigorous treatment Total field amplitude : Nonlinear polarization :  Neglect the 2nd order terms of E3 and E4

8. Wave equation : where, (1) Pump waves, A1 and A2 (slow varying approximation) # Each wave shifts the phase of the other wave by twice as much as it shift its own phase # Since only the phase of the pump waves are affected by nonlinear coupling, the quantities |A1|2 and |A2|2 are spatially invariant, and hence the k1 and k2 are in fact constant Solution : (6.1.15) : Nonlinear polarization responsible for producing the phase conjugate wave varies spatially. Therefore, two pump beams should have equal intensities :

9. (2) Signal (A3) and conjugate waves (A4) where, put,

10. Solution : ( conjugate wave at z=L is zero) i) : amplification ii) : depends on (can exceed 100%  pump wave energy)

11. Processes of degenerated four-wave mixing : One photon from each of the pump waves is annihilated and one photon is added to each of the signal and conjugate waves  Amplification of A3 and over 100% conversion of A4/A3are possible one photon transition two photon transition wave-vectors

12. Experimental set-ups

13. 17.5 Optical Resonator with Phase Conjugate Reflectors (A. Yariv) # The self-consistence condition is satisfied automatically every two round trips.  The phase conjugate resonator is stable regardless of the radius of curvature R of the mirror and the spacing l.

14. 17.6 The ABCD Formalism of Phase Conjugate Optical Resonator The wave incident upon the PCM : where, Reflected conjugate wave : By comparing the ABCD law for ordinary optical elements, Ray transfer matrix for the PCM mirror

15. ABCD law at any plane following the PCM : Example) Matrix after one round trip : Matrix after two round trip : : Self-consistence condition is satisfied automatically every two round trips

16. 17.7 Dynamic Distortion Correction within a Laser Resonator Phase conjugate resonator Distortion corrected beam

17. 17.8 Holographic Analogs of Phase Conjugate Optics 1) Holography reading recording

18. 2) Phase conjugate optics Holography by phase conjugation - Real time processing (no developing process) - Distortion free image transmission

19. 17.9 Imaging through a Distorted Medium Distortion free transmission (A2)

20. 6.2 Self-Focusing of Light Gaussian beam :

21. Self-Trapping : Beam spread due to diffraction is precisely compensated by the contraction due to self-focusing Simple model for self-trapping Critical angle for total internal reflection :

22. A laser beam of diameter d will contain rays within a cone whose maximum angular extent is of the order of magnitude of diffraction angle ; So, the condition for self-trapping : Critical laser power : Ex) CS2, n2=2.6x10-14 cm2/W, n0=1.7, l=1mm  Pcr = 33 kW # Independent of the beam diameter

23. 2w0 zf Simple model of self-focusing : total power where, where, : critical angle

24. 6.3 Optical Bistability : Two different output intensities for a given input intensity  Switch in optical computing and in optical computing Bistability in a nonlinear medium inside of a Fabry-Perot resonator (6.3.3) Intensity reflectance and transmittance : where, : amplitude reflectance and transmittance

25. 1) Absorptive Bistability In the case when only the absorption coefficient depends nonlinearly on the field intensity, at the resonance condition, Assume, Introducing the dimensionless parameter C, (6.3.7)

26. Assume the absorption coefficient obeys the relation valid for a two-level saturable absorber ; Intracavity intensity : where, (6.3.7)

27. 2) Dispersive Bistability In the case when only the refractive index depends nonlinearly on the field intensity, where, (6.3.3)  : linear phase shift : nonlinear phase shift Similarly as before,