36. Nonlinear optics: χ(2) processes

Transcription

1 36. Nonlinear optics: χ() processes The wave equation with nonlinearity Second-harmonic generation: making blue light from red light approximations: SVEA, zero pump depletion phase matching quasi-phase matching surface SHG When is it necessary to think about χ(3)?

2 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 z c t dt 0 NL NL P = ε χ E + χ E + () (3) In order for this to make sense, this series must converge. As a result, we must assume that So the most important nonlinear term is the nd order term: the one involving χ (). To simply the problem, let s ignore all the other terms. () (3) (4) χ >> χ >> χ >> χ (5)...

3 The wave equation with χ () nonlinearity So the wave equation can be written as: E n E = µ P z c t dt () 0 where the nd order polarization is given by: = ε 0 χ incident P t E As we saw in the last lecture, there are several non-linear processes that can occur, even if we restrict ourselves to χ (). Pick one particularly interesting one: second harmonic generation (SHG) of a single incident wave at frequency.

4 Second Harmonic Generation: SHG χ () z = 0 z = L In this process, we imagine that one laser at frequency (the fundamental ) 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

5 Describing the nd harmonic wave We are interested in the behavior of the field that oscillates at ; that is, the nd harmonic. We can assume that this field is of the form: jk z j t E z, t = A z e + c. c. where we require that the amplitude A (z) is slowly varying, and also that it vanishes at the input facet of the nonlinear medium: ( z ) A = 0 = 0 Furthermore, the wave vector of this wave is related to the refractive index of the nonlinear medium at frequency : k = Our goal is to determine A (z). n ( ) c

6 What equation must the nd harmonic obey? The nd harmonic wave must obey the wave equation, of course. ( ) E n E P z c t dt () = µ 0 As we have seen, the nd-order polarization results from the field at frequency - the fundamental. Putting in the spatial dependence explicitly: 0 ( j t jk z ) P t ε χ A e + = the amplitude of the incident field (the one at frequency ) Squaring: P t A e the wave vector here is the k of the incident field: k j[ k z t] = ε 0 χ nd -order polarization proportional to square of incident field = n ( ) c

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

8 Solving the wave equation in second order The nonlinear wave equation becomes: A 4χ jk = A z e e z c jk z jk z At this point, we could find a similar first-order differential equation for A, 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, A does not decrease very much with increasing z. A is independent of z. In this case, we can easily integrate both sides of this equation.

9 Integrate both sides A z z j χ = z ' k 0 c 0 [ ' '] j k z k z dz ' A e dz ' Define the 'phase mismatch' We can do the integral on the right side: z 0 Thus we ve arrived at a result! This is just A (z). 1 ' = 1 j k j k z ' j k z e dz e A z χ A () k = k k [ j kz] exp 1 k Note, this is just: π π n n λ λ 4π = λ ( n n )

10 The solution The intensity of the second harmonic radiation is proportional to A. sin k z I z A z I I z I z sin δ δ The intensity of the nd harmonic is proportional to the square of the intensity of the fundamental wave. And the square of the propagation distance z. k where δ = k z It also depends sensitively on δ, 0 the product of k and z/ δ δ = dimensionless phase mismatch sin δ δ

11 Phase matching for a χ () process To summarize: I z I z sinc δ where δ = k z SVEA and zero-depletion approximations give lowest order solution. Intensity of SHG radiation is proportional to the square of the input intensity. In the limit δ << 1, intensity of SHG radiation grows quadratically with propagation distance. Intensity of SHG is very sensitive to phase mismatch - maximum when k = 0 π z = n λ input ( n ) δ sin δ δ For example, how much does the SHG intensity drop if δ = 1? If δ = 1, then sinc (δ) = The condition δ < 1 corresponds to k < If the SHG medium is too thick for a given k, conversion efficiency suffers. L

12 What does phase matching mean? When k = 0, this means that n() = n(). The phase velocity of the input and the nd harmonic are equal. λ = λ. phase-matched: amplitude The two waves maintain the same relative phase as they propagate. not phase-matched: propagation distance When k is not zero, the phase velocity of the fundamental and nd harmonic are different, and λ λ. As z increases, the nd harmonic wave walks out of phase with the input wave. amplitude propagation distance The condition k L << 1 ensures that the two waves don t walk too far out of phase with each other before reaching the end of the SHG crystal.

13 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

14 SHG illustration optic axis Poynting Example of matching n() and n() in a nonlinear medium: 1.7 refractive indices for BBO e-ray θ o-ray vector refractive index n o n e for θ =.78º n e for θ = 0º For λ = 1064 nm, at this angle, n o () = n e () and thus k = 0. What if we changed the angle slightly? For example: 3º nm 1064 nm wavelength (µm) Then n o () is unchanged. But n e () = And thus: 4π -1 k = ( n n ) = 4150 m λ sin δ For a crystal of thickness = 1 mm: δ = k z =.1 and so = 0.18 δ

15 What if the phase matching is not perfect? intensity intensity of the fundamental (decreasing?) I z I z intensity of the nd harmonic (increasing quadratically if k = 0) ( k z ) ( k z ) sin 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 = 0 (quadratic) intensity decreasing peak signal with increasing k SHG crystal propagation distance

16 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 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 by π. Instead of cancelling out as it propagates beyond L coh, E would be further enhanced. In some cases, we can control the sign of χ () by changing the crystal structure. LiNbO 3

17 Quasi-phase matching Flipping the sign of χ () once each coherence length is known as quasi-phase matching. It has 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

18 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 = 1 + 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

19 Do we ever worry about χ (3)? P( t) [...] (1) () (3) 3 χ E t + χ E t + χ E = ε t 0 + If the power series is to converge, then χ (3) << χ () So when are χ (3) effects important? Usually, χ (3) is only important when χ () is equal to zero. It is easy to argue that χ () is zero most of the time

20 Symmetry considerations P () = ε 0 χ () E Consider a medium which exhibits inversion symmetry. many crystalline materials including all cubic crystals any amorphous material (glassy solid, liquid, gas) In a material like that, reversing the sign of E must also reverse the sign of the induced polarization. P () = ε 0 χ () [ E] P () = P () can be true only if χ () = 0, as well as χ (4), χ (6), etc. In this case, the largest non-linear effect is χ (3). Next lecture: χ (3) effects