Nonlinear Optics

1. Nonlinear Optics Introduction NLO effects have been observed since 19thC Pockels’ effect Kerr effect High fields associated with laser became available in the 1960s and gave rise to many new NLO effects second harmonic generation (SHG) third harmonic generation (THG) stimulated Raman scattering self-focussing In NLO we are concerned with the effects that the light itself induces as it propagates through the medium In linear optics the light is deflected or delayed but its frequency (wavelength) is unchanged We should remark that in order to observe nonlinear phenomena we need very intense light, that explains way nonlinear phenomena's where after the laser invention

2. The nonlinear response Consider an electronic amplifier Response is given by where x is the input signal and is the linear gain of the system Applied field is given by Output is given by The output is a faithful representation of the input When the applied voltage becomes large the amplifier become nonlinear and the output becomes distorted x=Vcos(wt)

4. Nonlinear response • Actual response can be written as • we have a cubic distortion • the output now becomes • Trigonometric identity gives • The output is thus given by • Thus a small cubic nonlinearity gives rise to a modified response at w but also generates a new signal at 3w

5. Optical nonlinearity Model of medium depicts charge clouds surrounding nucleus Applied field distorts the cloud and displaces the electron Compare with a mass on a spring Separation of charges gives rise to a dipole moment Dipole moment per unit volume is called the polarization F=Kx

7. Linear polarization • Consider only one dimension • can be written as Polarization • is the linear susceptibility • This describes linear propagation giving rise to • speed of propagation through the medium (real part) • absorption in the medium (imaginary part) • It can be shown that • where n is the refractive index of the medium

8. Nonlinear polarization • Expand the polarization as a power series in E to gives • where is the i-th order nonlinear susceptibilities • in order for the series to converge we must have • symmetry arguments can be used to show that for isotropic materials even orders are zero

9. Harmonic terms in the polarisation • Apply an electric field at frequency w • Higher order terms give rise to components at frequency w,2w,3w etc as we saw in the case of the electronic amplifier • Gather terms at the harmonics to get • Harmonic components in polarisation give rise to fields at harmonic frequencies • Consider the situation of an applied optical at frequency w and DC field • Higher order terms give cross-products

11. phenomena • linear electro-optic effect discovered by Pockels in 1883 • modification of the refractive index due to an applied DC electric field • used for optical switching • used for phase modulation of light • applied optical field gives rise to a component of the polarization at DC • static voltage appears across the sample which is proportional to the applied optical field • polarizations generated at frequency 2w • an optical field is generated at 2w • second harmonic generation • light at 1000nm will generate SHG at 500nm!

12. phenomena • quadratic eletro-optic effect • also known as the Kerr effect discovered in 1883 • occurs in isotropic media such as gases and liquids as well as solids • not as widely used as the Pockels effect due to quadratic dependence with applied field • not linear • much higher voltage needed • This is actually SHG • symmetry imposes the constraint that is zero for isotropic materials • SHG not possible in isotropic materials • applied electric field breaks the symmetry making SHG possible

13. phenomena optical (or AC Kerr ) effect • third order susceptibility permits a term at w caused by light a frequency w looks like a refractive index which depends on the optical field strength • gives rise to self-focussing and self-phase modulation • third harmonic generation • similar to SHG but is allowed in all materials including isotropic materials

14. SECOND HARMONIC GENERATION

17. What will happen if we apply more then one harmony?

18. Sum and difference-Frequency Generation

19. To summarize

21. Example for an experiment of DFG, throw this device we can detect the existent of w1.

22. Third order susceptibility case Inserting this into the polarization will give for example terms like

25. Symmetry of • In general is a tensor with 9 elements • Many elements will typically be zero • In centrosymmetric crystals • Consider the term • Since the polarization is generated by theapplied field we would expect it to be reversed when the field is reversed • Reversing E in the above leaves the polarization unchanged thus must vanish • We normally represent the elements of the nonlinear tensor as a 3x6 matrix

26. Two symmetry conditions on • The coefficients will remain the same under a permutation of indices provided the frequencies are also permuted. Thus • Kleinman’s conjecture ( Kleinman’s symmetry) • applies to lossless media • a good approximation for most relevant materials • It states that d is independent of frequency and is independent of the permutation of indices

27. Contracted notation for d • Express x,y,z as 1,2,3 • express each component of d as • the 3x6 matrix becomes

28. Nonlinear susceptibility of anharmonic oscillator We’ll try to describe the atom as anharmonic oscillator, where x is the electron position Dumping force Remark- centrocymetric crystals are characterized by U(x)=U(-x)

29. We will guess a solution We develop x in as a perturbation series Equating terms will give the following equations

30. We can see that the first equation is just a linear unhomorganic equation of an harmonic oscillator with the solution i where Now we can insert our solution to the second equation and That will give us other harmonies (SHG DfG ect) for example will take SHG: After inserting the solution to the first equation we can see that It is again unhomogenic harmonic oscillator.

31. We’ll see we get the same kind of solution And in the same manner one can derive x for the other harmonies for example:

32. Now we by definition of the dipole moment and x we can find the Susceptibility In the same manner we can deal with the SHG We can find in the same way other values of the second Order susceptibility (for different harmonies)

33. Maxwell equations When we assume And we also know Manipulating these equations can lead us to a wave equation

34. We’ll define

35. That will give us separated wave equations So we saw that manipulating the Maxwell equations, we Can derive wave equations and see that the polarization Is the source.

36. Coupled wave equation for sum frequency We’ll apply the field And after the media we’ll have the harmony

37. Inserting to the wave equation will give

38. Linearization of the equation Assuming that We can derive

39. Linearization of the equation We can also find the equations for the two other amplitude

40. Phase matching The solution for the amplitude equation is the following integral Thus will bring us to the phasing condition defining

42. sources • A slide show , Allister Ferguson http://phys.strath.ac.uk/12-370/index.htm • Nonlinear Optics 2nd, R.W.Boyd, Academic Press • Quantum electronics 3rd , Amnon Yariv