1. PHYS 408 Applied Optics (Lecture 13)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
Gaussian beams have various mathematical representations, but fundamentally are characterized by only two independent parameters (assuming a well-defined coordinate system).
Can choose the independent parameters from (R(z), w(z), z0, w0, l, etc.), but once you know those two independent parameters, you can define the Gaussian function everywhere in space, in a given reference frame. If you don’t know the reference frame, then there is an additional independent parameter involved.
The radius of curvature of the Gaussian wavefronts is infinite at z=0 and z=+ infinity and – infinity, and is a minimum (most curved) at z= z0, where z0 is the Rayleigh length.
They are only useful in practice for paraxial beams that do not diverge “too much”

3. The q parameterization
So, along with q, what do you need to know to specify the Gaussian? (Ans: k or lambda)

4. The real and imaginary parts of 1/q
If you knew R and W at some z value, do you know everything about the beam, everywhere else?
Must be yes since only two parameters…algorithm for finding lambda and z_0? (only unknown in q(z), since know z R and w, is lambda…solve for lambda, use imaginary part of q to get z_0

5. Follow q through a paraxial optical system

6. The ABCD Matrix: Ray Optics
Fundamentally based on Fermat’s Principle, which is?

7. Fermat’s Principle
Sketch refraction example…Fermat’s principle used to “derive” reflection and refraction angles, of “rays”, then more complicated geometrical surfaces are dealt with using the rays.

8. Examples
Sketch how Fermat’s principle would be used to get the refraction angle

9. Examples
Tell them to work out the single interface themselves at home, but give it, and the free space propagation Matrix, derive the thin lens result in class.

10. Examples (reflective surfaces)

11. Gaussian Example #1 (free space propagation)
Get them to work it out. Physically interpret the answer.
Emphasize that this is a “single Gaussian” defined everywhere in space by two parameters, this just “propagates” the q or 1/q parameters

12. Gaussian Example #2 (effect of a thin lens)
From the sketch, what are q1 and 1/ q1 at entrance to lens?
What is the relevant ABCD matrix to propagate just through the lens?
See handwritten notes: emphasize that now dealing with the transformation between two distinct Gaussian beams (through the lens). Extend both with red pen.
q1=0-I z_{01};
1/q1=0+(i * lambda/pi/w{01}^2);
Might guess that q2 there ~ -f – i* z_{02}, where z_{02} not obvious, or
1/q2 there ~ 1/R_2(-f) + i * lambda/pi/w_{01}^2, where R_{2(-f) not obvious.
Need to use ABCD matrix elements to obtain the relationships between the output Gaussian and the input Gaussian, and f.
What are q2 and 1/ q2 just after the lens?