Frequency Doubling and Second Order Nonlinear Optics


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1 Frequency Doubling and Second Order Nonlinear Optics Paul M. Petersen DTU Fotonik, Risø campus Technical University of Denmark, Denmark (
2 Outline of the talk The first observation of second harmonic generation Nonlinear optical effects in solids The nonlinear wave equation (formulation of the interaction in nonlinear materials) Second order nonlinear optics Contracted notation for the second order susceptibility Three wave mixing Frequency doubling Phase matching in materials with birefringence
3 The first observation of second harmonic generation Second harmonic generation (SHG) was first demonstrated by P. A. Franken et al. at the University of Michigan in They focused a ruby laser with a wavelength of 694 nm into a quartz sample. They sent the output light through a spectrometer and recording the spectrum on photographic paper, which indicated the production of light at 347 nm. (Note that the SHG spot does not appear in the publication!)
4 Conversion of red lasers into blue lasers
5 Generation of new waves in a nonlinear medium Light interacts with the nonlinear medium. The incident beams are modified and new beams are generated.
6 Optical Polarisation induced in Solids The optical electrical field induces electrical dipole moments in the nonlinear material. The induced dipoles modify the incident beam and for strong optical fields leads to the generation of new kvectors and new frequencies. P is the dipole moment per unit volume. The optical field vector The induced dipoles represents accelerating charged particles that radiates electromagnetic radiation perpendicular to the acceleration vector.
7 Second Order Nonlinear Polarization The relation between the induced optical polarisation in the material and the electric field. a) In a linear material b) In a crystal without inversion symmetry. The optical polarisation induced by an applied sinussoidal electric field with frequency ω. The polarisation consists of three contributions with frequencies ω, ω and 0 (dcterm)
8 Nonlinear Optical Effects in Solids The nonlinear optical effects are described by the polarization: (1) () (3) ε ε χ χ χ D = E + P, P( E) = ( E + E E + 4 E E E + ) 0 i 0 ij j ijk j k ijkl j k l Linear optics is described by χ ij (1) : The optical properties are independent of the light intensity The frequency of light does not change due to interaction with light The principle of superposion is valid Nonlinear optics is described by χ ijk () and χ ijkl (3) Light can interact with light The principle of superposion is not valid New waves with new kvectors and new frequencies are generated
9 Nonlinear Optical Effects in Solids The nonlinear optical effects are described by the polarization: D = ε E + P P E = ε χ E + χ E E + χ E E E + (1) () (3) 0, i ( ) ( 0 ij j ijk j k 4 ijkl j k l ) χ (1) ij describes the linear 1. order optical properties: absorption (α) and refraction (n) χ () ijk describes the. order nonlinearities: frequency doubling, electrooptic effect, and parametric oscillation, etc. χ (3) ijkl describes 3. order nonlinearities: quadratic kerr effect, intensitydependent refractive index, fourwave mixing, selffocusing, etc.
10 Quantum mechanical determination of the susceptibility χ The material response is given by the polarization: iωt iωt P( t) = ½ ε E{ χ( ω) e + χ( ω) e } 0 (1) The susceptibility χ(ω) may be determined by calculating P(t) quantum mechanically and then determine χ(ω) by comparison with Eq.(1). The macroscopic polarization with N atoms in a volume V is: P( t) = Nd( t) / V Where the electric dipole moment is ( ) = Ψ( ) Ψ( ) 3 z, X= x j j=1 d t t ex t d r
11 Formulation of the interaction in nonlinear materials The Maxwell equations: D H H = J +, E= µ 0, t t D= ε E + P, J = σ E lead to: the nonlinear wave equation The nonlinear polarization term P NL leads to the generation of new waves with new kvectors, polarization, and frequencies: For a new wave: 0 NL E E P E = µ 0σ + µ 0ε0ε + µ 0 t t t E z t Ae c c i( ωit ki z) i (, ) = ½ i +.. the nonlinear wave equation reduces (SVEAapproximation) to: dai 1 µ µ e + c. c. = σ Ae + c. c. + i dz i( ωit ki z) 0 i( ωit kiz) 0 1 i i ωi ε 0εi ε 0εi P t NL
12 The nonlinear wave equation dai 1 µ µ e + c. c. = σ Ae + c. c. + i dz i( ωit ki z) 0 i( ωit ki z) 0 1 i i ωi ε 0εi ε 0εi P t NL This is the fundamental equation that governs the generation of a i( ωit kiz) new wave E ( z, t) = ½ Ae + c. c. i i
13 Second Order Nonlinear Optics The interaction between light and matter is given by: where P E E E E E E E (1) () (3) i ( ) = ( ε 0χij j + χijk j k + 4 χijkl j k l + ) P ( ), E = χ E E () i NL ijk j k describes the second order nonlinearities. The notation χ = d () ijk ijk is often used in the literature where d ijk is a third order tensor. with 7 elements.
14 Contracted Notation for the Second Order Susceptibility P ( ), E = d E E i NL ijk j k where χ = d () ijk ijk since d ijk = d ikj The third order tensor may be represented by a (3x6) matrix: d with 18 elements. il d11 K d16 = M O M d31 a L 36
15 Generation of new waves using second order nonlinearities Sum frequency generation Second harmonic generation ω 1, ω ω 1 + ω Nonlinear crystal ω, ω Difference frequency generation Nonlinear crystal ω Optical parametric oscillator (OPO) Nonlinear crystal ω 1, ω ω 1 ω Nonlinear ω crystal ω pump signal, ω idler M M
16 Three Wave Mixing Two waves E(ω 1 ) and E(ω ) may due to the interaction with the second order nonlinear polarization generate a third beam E(ω 3 ) ω 1, ω ω 1, ω, ω 3 Nonlinear crystal The three waves are given by: E A e c c E A e c c E A e c c i ( ω1t + k1z ) i ( ωt+ k z ) i( ω3t + k3z) ( ω ) = ½ +.., ( ω ) = ½ +.., ( ω ) = ½ where we have assumed all waves to be polarized in the same direction. For the three waves the nonlinear wave Equation reduces to: da1 µ σ c iω µ αdc = A A A e dz n n * da µ σ c iω µ αdc = A A A e dz n n 0 0 * 3 1 da µ σ c iω µ αdc = A A A e dz n n where k=k 3 k 1 k, α= ( δ ω1 ω )/, and where we have used (δ ω1 ω =1 for ω 1 = ω and δ ω1 ω =0 for ω 1 #ω ) i kz i kz i kz ' The fundamental equations that describes second order nonlinear interaction. µ 0 µ 0c ε ε = n 0 i 3
17 Frequency doubling ω 1 = ω, ω = ω ω, ω 3 =ω Nonlinear crystal The fundamental equation da c i dc = µ σ A ω µ α A A e dz n n i kz reduces to: da( ω) dz iωµ 0dc = n( ω) A ( ω) e i kz where absorption is neglected. If we assume that A 1 =A =A(ω) is undepleted we obtain the second harmonic intensity: 3 3 µ c 0 ω d sin( kl / ) = L I ω I( ω) ( ) n ( ω) n( ω) kl / where L is the length of the nonlinear crystal.
18 Phase matching in materials with birefringence Strong SHG signals are obtained when : ω k=k3k1k =k3k 1 = ( n( ω) n( ω)) = 0 c Phase matching condition It was suggested independently by Giordmaine(ref.1) and Maker et al(ref.)) that index matching may be obtained in birefractive crystals by correct choice of polarisation and angle θ of propagation. The intersection defines the propagation direction. In Ref. phase matching in KDP with an 300fold increase of blue light intensity was observed in 196 Ref.1: J.A.Giordmaine, Phys. Rev. Letters, 8, 19(196); Ref: P. Maker et. al, Phys. Rev. Letters, 8, 1(196),
19 Type I and II phase matching Type I phase matching: The refractive index of an extraordinary beam is given by: ne ( θ ) = n0 Phase matching is obtained in negative uniaxial crystals (n e <n 0 ) when sin e n e n n o e θ + n cos ( ω, θ ) = n m θ 0 Type II phase matching: In this case the fundamental beam ω consist of both an ordinary beam and an extraordinary beam and in this configuration the beam at ω fulfils the condition: n ( θ, ω) = ½( n ( θ, ω) + n ( ω)) e m e m and the frequency doubled beam will be an extraordinary beam 0
20 Suggested further reading: Nonlinear Optics by R. W. Boyd Academic Press Optical Electronics in Modern Communications by A. Yariv, Oxford University Press The Principles of Nonlinear Optics by Y. R. Shen, Wiley
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