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

Nonlinear Optics Lab. Hanyang Univ. 6.1 Optical Phase Conjugation : Generation of a time-reversed wavefront Signal wave : Phase conjugate wave : where, : amplitude reflection coefficient

Nonlinear Optics Lab. Hanyang Univ. 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 [  -phase shift] ii) In reflection from PCM [  -phase shift & y component : i  /2  -i  /2 (-  : delayed)] 2) : The wavefront is reversed 3) : The incident wave is reflected back into its direction of incidence

Nonlinear Optics Lab. Hanyang Univ. 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

Nonlinear Optics Lab. Hanyang Univ.

Phase Conjugation by Degenerated Four-Wave Mixing 1) Qualitative understanding Four interacting waves : Nonlinear polarization : Counter-propagating waves) New wave (k 4 ) source term ! So, A 4 is proportional to A 3 * (complex conjugate of A 3 ) and its propagation direction is –k 3 (in the case of perfect phase matching)

Nonlinear Optics Lab. Hanyang Univ. 2) Rigorous treatment Total field amplitude : Nonlinear polarization :  Neglect the 2 nd order terms of E 3 and E 4

Nonlinear Optics Lab. Hanyang Univ. Wave equation : where, (1) Pump waves, A 1 and A 2 (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 |A 1 | 2 and |A 2 | 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 :

Nonlinear Optics Lab. Hanyang Univ. (2) Signal (   ) and conjugate waves ( A 4 ) where, put,

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

Nonlinear Optics Lab. Hanyang Univ. 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 one photon transitiontwo photon transitionwave-vectors  Amplification of A 3 and over 100% conversion of A 4 /    are possible

Nonlinear Optics Lab. Hanyang Univ. Experimental set-ups

Nonlinear Optics Lab. Hanyang Univ 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.

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

Nonlinear Optics Lab. Hanyang Univ. 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

Nonlinear Optics Lab. Hanyang Univ Dynamic Distortion Correction within a Laser Resonator Phase conjugate resonator Distortion corrected beam

Nonlinear Optics Lab. Hanyang Univ Holographic Analogs of Phase Conjugate Optics 1) Holography recording reading

Nonlinear Optics Lab. Hanyang Univ. 2) Phase conjugate optics Holography by phase conjugation - Real time processing (no developing process) - Distortion free image transmission

Nonlinear Optics Lab. Hanyang Univ Imaging through a Distorted Medium Distortion free transmission (A 2 )

Nonlinear Optics Lab. Hanyang Univ. 6.2 Self-Focusing of Light Gaussian beam :

Nonlinear Optics Lab. Hanyang Univ. 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 :

Nonlinear Optics Lab. Hanyang Univ. 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 : # Independent of the beam diameter Ex) CS 2, n 2 =2.6x cm 2 /W, n 0 =1.7, =1  m  P cr = 33 kW

Nonlinear Optics Lab. Hanyang Univ. Simple model of self-focusing 2w 0 zfzf where,: critical angle where, : total power

Nonlinear Optics Lab. Hanyang Univ. 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 Intensity reflectance and transmittance : where,: amplitude reflectance and transmittance (6.3.3)

Nonlinear Optics Lab. Hanyang Univ. 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)

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

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

Nonlinear Optics Lab. Hanyang Univ.