Lecture 6. P ω ω ε χ ω ω ω ω E ω E ω (2) χ ω ω ω χ ω ω ω χ ω ω ω (2) (2) (2) (,, ) (,, ) (,, ) (2) (2) (2)

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1 Lecture 6 Symmetry Propertes of the Nonlnear Susceptblty Consder mutual nteracton of three waves: ω, ω, ω = ω + ω 3 ω = ω ω ; ω = ω ω 3 3 P ω ω ε ω ω ω ω E ω E ω n + m = 0 jk m + n, n, m j n k m jk nm Then need to determne sx tensors.,,,,,, ω ω ω ω ω ω ω ω ω jk 3 jk 3 jk 3 ω, ω, ω ω, ω, ω ω, ω, ω jk 3 jk 3 jk 3 and 6 addtonal terms where each frequency s negatve. Indces,j,k can ndependently take on values x,y,z. Each of tensors consst of 7 Cartesan components. Thus, 34 complex numbers to fully descrbe nteracton. Restrctons and smplfcatons relate components of => Need far fewer n practce.

2 . Realty of the Felds Nonlnear polarzaton of sum-frequency response for nput m, n Pɶ rɶ, t = Pɶ ω + ω exp[ ω + ω t] + Pɶ ω ω exp[ ω + ω t] Pɶ, s purely real rɶ t P ω ω = Pɶ ω + ω also n m n m n m n m n m n m E ω = E ω and E ω = E ω * * j n j n k m k m * ω ω 3 From and 3 t follows that: ω ω, ω, ω = ω + ω, ω, ω * jk n m n m jk n m n m 4. Intrnsc Permutaton Symmetry Dummy ndces j,k,m,n may be nterchanged: + ;, E E thus ω ω ω ω ω ω kj n m m n k m j n ω + ω, ω, ω = ω + ω, ω, ω jk n m n m kj n m m n Physcally, the order of the felds n products does not matter.

3 3. Symmetres for Lossless Meda If medum s lossless, then all components of We can then employ Full Permutaton Symmetry Usng the realty condton 4, rght hand sde becomes So that: jk are real-valued. e.g. classcal anharmonc oscllator when all appled frequences, sums, and dfferences are sgnfcantly dfferent then resonance frequency of system. All frequency arguments of nonlnear-susceptblty tensor can be freely nterchanged, as long as the correspondng Cartesan ndces are smultaneously nterchanged. ω = ω + ω = ω = ω ω jk 3 jk 3 ω = ω + ω * jk 3 ω = ω + ω = ω = ω + ω jk 3 jk 3 ω = ω + ω = ω = ω ω jk 3 kj 3 A general proof of full permutaton symmetry entals quantum-mechancal expresson for or from a consderaton of the feld energy densty. 5

4 4. Klenman s Symmetry If ω ω then 0 the lowest resonant frequency s ndependent of frequency. Therefore the system responds essentally nstantaneously, and n tme-doman: P ɶ t = E ɶ t and s a constant. Snce s not frequency dependent, we can therefore permute the ndces: = = = = = jk jk kj kj jk kj

5 Number of Independent Elements of jk ω ; ω, ω 3 34 numbers n general must be specfed. 3 due to realty of the physcal felds. due to ntrnsc permutaton symmetry. snce all elements are real, and full permutaton symmetry s vald. 7 are ndependent. 8 for second harmonc generaton. 0 when Klenman s symmetry s vald. Crystallne symmetres of the nonlnear materal can reduce ths number further.

6 Effectve Value of dd eff For a fxed geometry.e. fxed propagaton and polarzaton drecton t s possble to express the nonlnear polaratongvng rse to sum-frequency generaton, by means of the scalar relatonshp For SHG d eff P ω = 4 ε d E ω E ω 3 0 eff P ω ε d E ω = 4 0 eff where P = P and E = E d jk ω ω ω ω = jk jk s obtaned by evaluaton of the summaton n general formula d eff depends on the crystal class calculated for dfferent crystals and tabulated also depends on the whether two low-frequency waves have the same polarzaton type I condton or the polarzatons are orthogonal type II condton d eff