12. Nonlinear optics I

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1 1. Nonlinear optics I What are nonlinear-optical effects and why do they occur? Maxwell's equations in a medium Nonlinear-optical media Second-harmonic generation Conservation laws for photons ("Phasematching") Quasi-phase-matching 1

2 Nonlinear Optics can produce many exotic effects. Sending infrared light into a crystal yielded this display of green light: Nonlinear optics allows us to change the color of a light beam, to change its shape in space and time, to switch telecommunications systems, and to create the shortest events ever made by humans.

3 Why do nonlinear-optical effects occur? Recall that, in normal linear optics, a light wave acts on a molecule, which vibrates and then emits its own light wave that interferes with the original light wave. We can also imagine this process in terms of the molecular energy levels, using arrows for the photon energies: 3

4 Why do nonlinear-optical effects occur? (continued) Now, suppose the irradiance is high enough that many molecules are excited to the higher-energy state. This state can then act as the lower level for additional excitation. This yields vibrations at all frequencies corresponding to all energy differences between populated states. 4

5 Reminder: Maxwell's Equations in a Medium The induced polarization, P, contains the effect of the medium. The inhomogeneous wave equation (in one dimension): E 1 E = µ P x t dt c The polarization is usually proportional to the electric field: P = ε χe Then, the wave equation becomes: E 1 E = ε µ χ E x c t dt E ( 1+ χ ) E = x c t χ = unitless proportionality constant Recall, for example, in the forced oscillator model, we found: Ne E ( t) P( t) = m ω ω + iγω since 1 = ε µ or c 5

6 Reminder: Maxwell's Equations in a Medium E x ( 1+ χ ) E = t c But this is the same equation as the usual homogeneous equation, if we define a new constant c: And, we call the quantity 1+ χ the refractive index. So, we can describe light in a medium just like light in vacuum, as long as we take into account the refractive index correction. But this only worked because P was proportional to E ( 1 χ ) 1 + = c c What if it isn t? Then P is a non-linear function of E! 6

7 Maxwell's Equations in a Nonlinear Medium Nonlinear optics is what happens when the polarization is the result of higher-order terms in the field: P = ε χ E + χ E + χ E + = P + P (1) () (3) 3... Linear non linear Then the wave equation must look like this: x c t dt E n E P = µ non linear The linear term can be treated in the same way as before, giving rise to the refractive index. But the non-linear term is a problem E n E () ( ) (3) ( 3 = ε ) µ χ E + ε µ χ E + x c t dt dt Usually, χ (), χ (3), etc., are very small and can be ignored. But not if E is big 7

8 The effects of the non-linear terms What sort of effect does this non-linear term have? If we write the field as: then E( t) E exp( iωt) + E exp( iωt) * * ω E( t) E exp( i t) + E + E exp( iωt) terms that vary at a new frequency, the nd harmonic, ω! Nonlinearity can lead to the generation of new frequency components. This can be extremely useful: Frequency doubling crystal: 164 nm 53 nm 8

9 Sum and difference frequency generation Suppose there are two different-color beams present, not just one: E( t) = E exp( iω t) + E exp( iω t) + E exp( iω t) + E exp( iω t) * * Then E(t) has 16 terms: E( t) E exp( iω t) + E exp( iω t) * E exp( iω t) + E exp( iω t) * [ ω ω ] * * [ ω ω ] [ ω ω ] * * [ ω ω ] + E E exp( i + t) + E E exp( i + t) E E exp( i t) + E E exp( i t) E + E 1 nd harmonic of ω 1 nd harmonic of ω zero frequency - known as optical rectification sum frequency difference frequency This is an awful lot of processes - do they all occur simultaneously? Which one dominates (if any)? What determines the efficiency? 9

10 Complicated nonlinear-optical effects can occur. Nonlinear-optical processes are often referred to as: "N-wave-mixing processes" Emitted-light photon energy where N is the number of photons involved (including the emitted one). This cartoon illustrates a 6-wave mixing process. It would involve the χ (5) term in the wave equation. The more photons (i.e., the higher the order) the weaker the effect, however. Very-high-order effects can be seen, but they require very high irradiance, since usually χ () > χ (3) > χ (4) > χ (5) 1

11 Conservation laws for photons in nonlinear optics Energy must be conserved. Recall that the energy of a photon is. Thus: ħω ω1 + ω + ω3 ω4 + ω5 = ω Photon momentum must also be conserved. The momentum of a photon is, so: ħk k + k + k + k + k = k k k But is related to ω : So may not correspond to a light wave at frequency ω! Satisfying these two relations simultaneously is called "phase-matching." Usually, only one (or zero) of the many possible N-wave mixing processes can be phase-matched at a time. k π n = = λ nω c 11

12 Phase-matching: an example Consider the nd harmonic generation process: Energy conservation requires: ω in ω out Refractive index n nonlinear material Momentum conservation requires: k ( ω) + k ( ω) = k ( ω ) ( ω ) n( ω) ω Frequency ω red photons 1 blue photon ω ( ω) = n( ω) n c n( ω) = n( ω) ω c Unfortunately, dispersion prevents this from ever happening! 1

13 Phase-matching Second-Harmonic Generation using birefringence Birefringent materials have different refractive indices for different polarizations: the Ordinary and Extraordinary refractive indices! Using this, we can satisfy the phase-matching condition. For example: Use the extraordinary polarization for ω and the ordinary for ω: Refractive index n e n o n o ( ω) = n ( ω) e ω Frequency ω n e depends on propagation angle, so by rotating the birefringent crystal, we can tune the condition precisely by moving the red curve up and down relative to the blue curve. 13

14 Light created in real crystals Far from phase-matching: SHG crystal Input beam Output beam Closer to phase-matching: SHG crystal Input beam Output beam Note that SH beam is brighter as phase-matching is achieved. 14

15 Second-Harmonic Generation SHG KDP crystals at Lawrence Livermore National Laboratory These crystals convert as much as 8% of the input light to its second harmonic. Then additional crystals produce the third harmonic with similar efficiency! 15

16 Difference-Frequency Generation: Optical Parametric Generation, Amplification, Oscillation Difference-frequency generation takes many useful forms. ω 1 ω 3 ω = ω 3 ω 1 Parametric Down-Conversion (Difference-frequency generation) ω 1 ω 3 ω Optical Parametric Generation (OPG) "signal" "idler" By convention: ω signal > ω idler ω 1 ω 1 ω 1 ω 3 ω ω 3 ω Optical Parametric Amplification (OPA) mirror Optical Parametric Oscillation (OPO) All of these are χ () processes (three-wave mixing). mirror 16

17 Another nd -order process: Electro-optics Applying a voltage to a crystal changes its refractive indices and introduces birefringence. In a sense, this is sum-frequency generation with a beam of zero frequency (but not zero field!). Pockels cell V Polarizer If V =, the pulse polarization doesn t change. If V = V p, the pulse polarization switches to its orthogonal state. The Pockels effect can be described as a χ () nonlinear optical interaction, where E E(ω) E(ω = ). Sum frequency is at ω + = ω. 17

18 The wave equation with nonlinearity We have derived the wave equation in a medium, for the situation where the polarization is non-linear in E: linear optics where E n E P = µ x c t dt NL NL P = ε χ E + χ E + () (3) 3... Usually, χ >> χ () (3) In these cases, we neglect the third (and higher) orders. A good example: second harmonic generation 18

19 Second Harmonic Generation: SHG ω ω χ () ω z = z = L In this process, we imagine that one laser (at frequency ω) is used to illuminate a nonlinear medium. As this field propagates through the medium, its intensity will be depleted and the intensity of the nd harmonic wave (initially zero) will grow. intensity intensity of the fundamental intensity of the nd harmonic propagation distance z 19

20 Describing the nd harmonic wave We are interested in the behavior of the field that oscillates at ω; that is, the nd harmonic. We can assume that this field is of the form: ( ω ) E z t A z e e c c ik ω z i t, = +.. ω ω where we require that the amplitude A ω (z) is slowly varying, and also that it vanishes at the input facet of the nonlinear medium: ( z ) A = = ω Furthermore, the wave vector of this wave is related to the refractive index of the nonlinear medium at frequency ω: k ω = Our goal is to determine A ω (z). n ( ω ) ω c

21 What equation must the nd harmonic obey? The nd harmonic wave must obey the wave equation, of course. ( ω ) E n E P z c t dt () ω ω = µ As we have seen, the nd-order polarization results from the field at frequency ω - the fundamental. Putting in the spatial dependence explicitly: ( iω t + ik z ) ω P t = ε χ E e the amplitude of the incident field (the one at frequency ω) i[ k z t] ω ω P t = ε χ E e this is the k of the incident field: k ω = n ( ω ) ω c 1

22 Plugging in to the wave equation Plug our assumed forms for E ω (z,t) and P (), to find: jk k A n A e z z c A ω A ω ω + ω ω ω + ω χ = c ( ω ) ω ( ) ω i[ k z ωt] E e i k z t ω Slowly Varying Envelope Approximation (SVEA): A z << k A z ω ω ω So we neglect the second derivative of A ω.

23 Solving the wave equation in second order The nonlinear wave equation becomes: A ik E z e e z c ω 8χ ω ikω z ikω z ω = Now, we could find a similar first-order differential equation for E, and then solve the two coupled equations. But, instead of doing that, let s see if we can gain some physical insight by making another simplifying assumption: Assume: The incident field is not significantly depleted by the conversion process. That is, E does not decrease very much with increasing z. E is independent of z. In this case, we can easily integrate both sides of this equation. 3

24 Integrate both sides A z z ω 4iχ ω i k z k z = z ' k ωc [ ' '] ω ω dz ' E e dz ' Define the 'phase mismatch' ω We can do the integral on the right side: Thus we ve arrived at a result! z i k z ' i k z e dz e This is just A ω (z). A z E ω k = k k 1 ' = 1 j k [ i kz] exp 1 k ω Note, this is just: π π nω n λ λ in 4π = ω λ in in ( n n ) 4 ω ω

25 The solution The intensity of the second harmonic radiation is proportional to A ω. sin k z ω ω I z A z I sin = I z δ The intensity of the nd harmonic is proportional to the square of the intensity of the fundamental. δ k where δ = k z δ = dimensionless phase mismatch sin δ δ It also depends sensitively on the product of k and z δ 5

26 Phase matching for a χ () process To summarize: sin k z ω I z I z k z SVEA and zero-depletion approximations give lowest order solution. Intensity of SHG radiation is proportional to the square of the input intensity. Intensity of SHG radiation grows quadratically with propagation distance. Intensity of SHG is very sensitive to phase mismatch - maximum when k = δ sin δ δ SHG intensity is most efficient for δ < 1 If δ = 1, then sin δ/δ =.71. δ < 1 corresponds to k < If the SHG medium is too thick for a given k, conversion efficiency suffers. L 6

27 What does phase matching mean? When k =, this means that n(ω) = n(ω). The phase velocity of the fundamental and nd harmonic are equal. λ ω = λ ω. phase-matched: amplitude propagation distance When k is not zero, the phase velocity of the fundamental and nd harmonic are different, and λ ω λ ω. As z increases, the nd harmonic wave gets increasingly out of phase with the fundamental. not phase-matched: amplitude propagation distance This is why k L << 1 is the important condition to satisfy. 7

28 Materials and configurations for χ () NLO There are a number of materials commonly used for SHG or other frequency conversion effects based on χ (). KDP: potassium di-hydrogen phosphate BBO: beta-barium borate LiNbO 3 : lithium niobate etc. A non-linear crystal inside the laser cavity to produce UV light: LiNbO 3 crystals This is a VECSEL : a vertical external cavity surface emitting laser 8

29 SHG illustration Example of matching n(ω) and n(ω) in a nonlinear medium: 1.7 refractive indices for BBO e-ray optic axis θ o-ray k vector refractive index nm 164 nm wavelength (µm) n o n e for θ =.78º n e for θ = º For a crystal of thickness = 1 mm: δ = k z =.1 For λ = 164 nm, at this angle, n o (ω) = n e (ω) and thus k =. What if we changed the angle slightly? For example: 3º. Then n o (ω) is unchanged. But n e (ω) = And thus: 4π -1 k = ( nω nω ) = 415 m λ and so sin δ =.18 δ 9

30 What if the phase matching is not perfect? intensity intensity of the fundamental (decreasing?) intensity of the nd harmonic (increasing quadratically if k = ) ( k z ) ( k z ) sin ω I z I z propagation distance z If the phase mismatch is not precisely zero, then how does the second harmonic intensity behave? The SHG intensity oscillates as a function of propagation distance: k = (quadratic) ω intensity decreasing peak signal with increasing k SHG crystal propagation distance 3

31 Another way to boost the SHG efficiency SHG crystal Why does the signal oscillate? If phase matching condition is not perfect, then after a certain length (called the coherence length L coh ), the fundamental and nd harmonic walk out of phase with each other. At that point, the process reverses itself, and the fundamental grows while the ω beam diminishes. This process then oscillates. What if, at z = L coh, we could flip the sign of χ ()? This would change the phase of E ω by π. Instead of cancelling out as it propagates beyond L coh, E ω would be further enhanced. In some cases, we can control the sign of χ () by changing the crystal structure. LiNbO 3 31

32 Quasi-phase matching Flipping the sign of χ () once each coherence length is known as quasi-phase matching. It has recently become a critically important method for efficient second harmonic generation. (Length) phase matching quasiphase matching no phase matching The process of fabricating a material where the sign of χ () flips back and forth is known as periodic poling. A photo of PPLN: periodically poled lithium niobate 3

33 SHG at a surface Another method of minimizing δ = k z / : use a very small value of z. For example, at a surface or an interface. Ω = ω 1 + ω surface second harmonic generation - a very sensitive probe of surfaces (but very weak!) Applications: measuring the orientation of molecules at a liquid surface studying buried interfaces, e.g., silicon/insulator 33