1. Non-linear Effects
100mW of cw output from a package 3.8cm x 3.8 cm x 10cm.The device consists of a chip of 0.5mm of Nd:YV04 in contact with a 2mm KTP crystal; 500mW of laser output at 809nm is used to pump the device.

2. And inside the box…

3. Doubling and doubling again!

4. Non-linear Optics
The polarisation, P, can be described in terms of the susceptibility tensor, χ. We can include any non-linear response of the medium as shown below.
Note that in general the electric field and the polarisation need NOT be collinear.

5. Non-linear Effects: (Classification by order)

6. Anisotropic binding of an electron in a crystal
The springs have different stiffness for different directions of the electron’s displacement from its equilibrium position within the lattice.
The polarisation, and
therefore the refractive index, will be different in different directions

7. Index ellipsoid and the principal axes

8. Uniaxial or Biaxial? Biaxial crystal – two optical axes.
(b) Positive uniaxial crystal – one optical axis ne ≥ no
(c) Negative uniaxial crystal – one optical axis ne ≤ no

9. The Permittivity Tensor

10. The Index Ellipsoid
3D ellipsoid surface is given by:

11. The Electro-optic Ellipsoid
Taylor expand the refractive index in the electric field
Or in terms of r,
So,
With the identities,

12. Linear Electro-optic Tensor
The r-coefficients
are related to the
crystal symmetry
The 27-elements reduce to 18 because of invariance w.r.t. i,j interchange

13. Electro-optic tensor

14. Electro-optic effect Distortion of index ellipsoid caused by
The application of an electric field

15. Case of ADP and isomorphs
With a field applied in the z-direction the ellipsoid is distorted in the xy-plane
Rotation by 45° about the z-axis transforms to the (x´, y´, z´)
co-ordinate system

16. Rotation of Axes
If we take
θ=45 degrees

17. The ellipsoid referred to rotated axes

18. The transformed axes
So, given,
we can write,
i.e.,

19. Variable phase-plate With light polarised along the original
x-direction
For a half-wave plate:
(Note: the field
& propagation
directions are
the same here!)
Giving a half-wave
Voltage:

20. Amplitude modulation

21. Electric field & Displacement Vector

22. The wave-vector surface
The intercept of the
k-surface with each
plane xy, xz & yz
Consists of one circle
and one ellipse. The
surface is double suggesting there
are two possible values
for k for any direction
of the vector k. There are
two phase velocities
corresponding to two
orthogonal polarisations.
At the point P the two values are equal; this is the optical axis.

23. Poynting’s vector- S & Wave vector- k

24. Summary Linear optics: Non-linear optics: Linear electro-optic:
(Uniaxial crystal becomes biaxial when the field is applied)