Grading. Class attendance: (1 point/class) x 9 classes = 9 points maximum Homework: (10 points/hw) x 3 HW = 30 points maximum

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1 Grading Class attendance: (1 point/class) x 9 classes = 9 points maximum Homework: (10 points/hw) x 3 HW = 30 points maximum Maximum total = 39 points Pass if total >= 20 points Fail if total <20 points

2 Light propagation in a non-linear medium Maxwell s equations in a insulating material: D = 0, B = 0, E = 1, H = 1 c c Assume a non-magnetic (M = 0 B = H) If M 0 many interesting effects are possible (Kerr, Faraday, etc.). Set this aside. Assume a polarizable P 0 material (D = E + 4ππ) Take E = 0 inside the insulator Then, E = 2 E + E = 2 E and 2 E = 1 2 E 4π 2 P c 2 t 2 c 2 t 2 If P = 0 we get the free plane wave propagating with speed c If P 0 we have an interaction of light with the medium given by the term 4π 2 P c 2 t 2 Consider plane waves going along the z axis in the material so that the spatial dependence is on z only P α z, t = χ β α,β E β z, t + β,γ χ α,β,γ E β z, t E γ z, t + β,γ,δ χ αααα E β z, t E γ z, t E δ z, t + The non-linear terms in P describe the appearance of non-linear optical effects Typical values of the 1 st, 2 nd and 3 rd order susceptibility are χ ii χ 2 iii = 2d iii m V 3 χ iiii m V 2 Since most materials have dielectric breakdown for fields E 10 8 V, the expansion of P in powers of E m is justified

3 Second harmonic generation We assume non-resonant interactions in the following. This is usually the case, as we like to have a transparent medium to minimize absorption losses. Two-level resonant non-linear optics is at the basis of many remarkable effects (self-induced transparency, etc.), but we set this aside too. Take E β z, t = Ee βe ik 1z iω 1 t + c. c. Then, the non-linear term proportional to χ 2 makes the polarization acquire a spectral component at 2ω 1 which is P α z, t = E 2 χ ββ 2 e βe γe 2ik 1z 2iω 1 t + c. c This oscillating polarization will radiate at frequency 2ω 1, that is we will have light at twice the incident frequency This is called Second Harmonic Generation (there are also components of P α with zero frequency, that cause an effect called optical rectification ) When adding two waves we require that ω = ω ttt and k = k ttt The 2 nd condition may be called momentum conservation if we are interested in the photon representation. A more useful description is in the wave representation, where the 2 nd condition is called phase matching.

4 Phase matching Consider two sources of the SHG field, separated in space by a distance L Source S 1 starts earlier (let s say at time t) but its 2ω field has to travel a longer distance. Source S 2 starts later (at time t + T) but it is located closer to the detector. The phases of the SHG fields from S 1 and S 2 are φ 1 = 2ωω k 2ω z d z 1 and φ 2 = 2ω t + T k(2ω)(z d z 2 ), where z 1,2 are the position of the sources along the z axis Then, Δφ = φ 2 φ 1 = 2ωω k 2ω L and, with T = L c 2ωω c n ω n 2ω. This is a problem because n 2ω n ω > 0 in non-absorption ranges. n 2ω 2ω n ω and k 2ω =, we get Δφ = c Det 0 S 1 S 2 z Typical variations of the index of refraction with wave length are Δn = n 2ω n(ω) 10 2 Two SHG sources on the opposite ends of the crystal begin to interfere destructively when Δφ = π or at a thickness L c = cc = λ 0 800nn 20 μμ. For crystals longer than L 2ωΔn 4Δn c the SHG intensity will begin to decrease.

5 Birefringent crystals Phase matching is not possible, unless we refer to birefringent crystals, characterized by an anisotropic index of refraction in a special coordinate system with axes oriented along the crystal axes The dielectric constant of birefringent crystals is a tensor, which can be represented by a diagonal matrix ε ε = 0 ε ε 3 For simplicity, consider the case when ε 2 = ε 3, so that there are only two different values for the dielectric constant ( uniaxial crystals) This gives two indices of refraction for uniaxial crystals, which are n o = ε 1 and n e = ε 2 and are parameters that depend on ω, like all indices of refraction

6 Optics of birefringent crystals An illustration can be made of the variation of the index of refraction with the direction of a light beam in an uniaxial material The vertical axis is oriented along the uniaxial crystal optic axis and we consider a wave with wave vector k making an angle θ with this axis The indices of refraction the wave will experience depend on its polarization A wave polarized out-of-plane will have an index of refraction n o θ = n o, which does not depend on θ A wave polarized in-plane will have an index of refraction n e (θ), which depends on θ according to the relation 1 cos θ 2 = n 2 e (θ) n2 + o sin θ 2 n e 2. These quantities can be illustrated by the length of the intersection of k with a circle and by the length of the intersection with an ellipse. The illustration refers to a positive uniaxial crystal, where n e > n 0 (for example, TiO 2 ) If n e < n 0 we have a negative uniaxial crystal (the ellipse will fit inside the circle instead) (for instance, calcite or LBO) D e D o θ n o n e (θ) k Positive uniaxial crystal Exercise: find the direction of the optic axis in the calcite crystal. Hint: what is n e when θ = 0?

7 Birefringent crystals and phase matching We are looking for ways to make n 2ω n ω = 0 The parameters n o,e depend on ω just like for other materials and we can not obtain the above condition with either one alone However, because we now have two indices of refraction, we can make one of them at 2ω match the other at ω For instance, the sketch shows how to get phase matching n o 2ω = n e ω, in a positive uniaxial crystal when the wave vector is at a certain angle θ from the optic axis In practice, it is the crystal that is rotated (not k ) to achieve the phase matching condition The sketch shows how we can convert two e waves into one o wave, e + e o (called type I SHG) A second case is possible, where o + e o (type II SHG). For positive crystals, we must have an o wave at 2ω, because n o must catch up with the n e. For negative uniaxial crystals, the roles of the o and e waves are interchanged, so that type I SHG is o + o e and type II SHG is o + e e n o (2ω) In practice, we also have to know how to orient the polarization of the incident ω beam and what to expect of the polarization of the outgoing 2ω beam This is an example for LBO, which is a negative uniaxial crystal θ n e (ω) Polarization of the beam at ω for type I SHG (o wave) Polarization of the beam at ω for type II SHG (both e and o waves) Polarization of the beam at 2ω (e wave for both type I and II)

8 Walkoff Walkoff is the angular divergence of the o and e waves This is a problem in type II SHG, because this type combines two different waves. For a large conversion efficiency the two waves should follow the same path In a birefringent crystal, even though the wave vectors k o and k e are aligned, as shown by the parallel wave fronts in the sketch, the directions of the waves, giving the energy flow proportional to E H, are not because k e E e H e for the e wave Wavefronts of the o wave ρ k e k o E o H o Wavefronts of the e wave E e H e The angle ρ is given by the relation n 2 o tan(θ) = n 2 e tan(θ + ρ) Example: for calcite, n e = 1.486, n o =1.658 (it has a relatively small wavelength dependence in the visible, as you can see) Then, ρ = 0 if θ = 0 (this is a different way of understanding how the optic axis may be found) ρ = AAAA n o 2 n2 TTT(θ) θ = 6.02 o if θ = 50 o (you can see that the two beams are separated by e different angles as the crystal is rotated)

9 Optical mixing Non-linear effects can be observed with CW beams The NL effects are much more easily observable at the high peak intensities of pulsed lasers In practice, the efficiencies of the NL processes increase at high intensities. For instance, the efficiency of SHG can approach 90% range at I 5 GG/cm 2. This has lead to wide application of non-linear effects in ultrafast laser science and technology We started with P α z, t = χ 1 2 β α,β E β z, t + β,γ χ α,β,γ E β z, t E γ z, t If we now take E β z, t = E 1 e βe ik 1z iω 1 t + E 2 e βe ik 2z iω 2 t + c. c. (two fields of different frequencies), we obtain additional cross terms in P α (z, t) at various frequencies For instance, P α ω 1 +ω 2 χ 2 E 1 ω 1 E 2 ω 2 (sum frequency generation). If ω 1 = ω 2 we obtain SHG. P α ω 1 ω 2 χ 2 E 1 ω 1 E 2 ω 2 (difference frequency generation). This is applied in optical parametric oscillators and amplifiers in tunable fs lasers. + Third-order processes P α z, t = χ β α,β E β z, t + β,γ χ α,β,γ E β z, t E γ z, t + β,γ,δ χ αααα E β z, t E γ z, t E δ z, t Examples: optical Kerr effect, 4-wave mixing + Reminder: all are non-resonant processes in technological applications, because we require little absorption. Resonances may increase χ n by many orders of magnitudes and scientists like them, but they also introduce absorption which is why they are avoided in practice.