1. PHYS 408 Applied Optics (Lecture 21)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Logistics
Lab tour?
When?
Review session?

3. Quick review of key points from last lecture
The key to diffraction is realizing that if you do a 2D Fourier Transform of the field distribution in some x-y plane, and assume that you can relate that to a forward, z-propagating overall field, then each of the 2D Fourier components, maps uniquely onto a 3D planewave propagating normal to the plane.
The wavevector in the propagation direction of each of these 3D planewaves is given by , and the relative amplitudes of each of these 3D planewaves is given directly by the 2D Fourier amplitude of the corresponding in-plane component of the field in the aperture.
In the paraxial limit, each of these 3D plane waves, propagates at an angle

4. Today:
How a thin lens renders the Fourier transform of the field at an input plane at its focal plane
How diffraction limits the resolution one can achieve with light beams
A half-hour crash course on polarization!
Given the last sentence in the review, it is sort of “obvious” that the lens basically focusses each k// component of the field on the screen a distance f away, at a point x=f kx/k_0 (sketch in-coming plane wave to lens at an angle kx/k_0, the lens doesn’t alter direction of plane wave, just focusses it at a distance f away.

5. Effect of a lens
Why is this a bad diagram?

6. Physics approach
What does a lens do with a plane wave incident at some (small) angle q?
What fields are emanating from the f(x,y) plane?
So within a phase factor, the field at some location in the focal plane of the lens is proportional to what?
Sketch plane wave and lens on board.
Use 2 rays from ray optics to sort of figure it out…what principle would you use to prove all rays end up there?
So, regardless of a phase factor, what is the intensity at each point in the focal plane of the lens proportional to?

7. Rigorous approach
Show that a lens focusses a plane wave incident at some (small) angle q, to a point at fq from the optic axis on the screen?
Get them to say think through the 3 steps involved: i) field at output of lens, ii) it’s FT, and iii) propagate it a distance f.
Go through equations one by one, getting them to describe in words what each equation represents.
Pause after 2nd equation and get them to formulate the propagation integral, then show handwritten notes (page 3 of Lens Fourier Transform II.jnt) l- l+ s
Hint: propagate a phase front at the input to the lens of to the focal plane, knowing the transmission function of the thin lens is

8. Next, keep track of the relative phases of the plane waves that have propagated from input plane a distance d1 to the thin lens
Show pages 4 and 5 of .jnt file
Point is, simply by inspection, can see that each Fourier component of the aperture field just gets multiplied by a k// independent phase factor, and a k// dependent, quadratic phase factor.
Since we know how propagates through lens and to screen, just have to substitute so as to pick up the correct weighting for each Fourier Component.

9. …and the answer is:
See handwritten notes: note that the argument of the FT function F is kx, ky so kx=omeg/c x/f, so x/f=kx/omega (the angles are the same).

10. How does diffraction influence image resolution?
Think back to our thought-experiment:
If make ideal image out of only the propagating plane waves (excluding evanescent waves), what is the wavelength of the highest frequency spatial frequency component (wavevector)?
Max in-plane k//=k_0; E~e^{ik_0x}+e^{-ik_0x} (standing wave); |E|^2~cos{k_0 x}^2~ 1-sin(2 k_0 x); So spacing of peaks when 2 k_0 \delta x= 2 pi; so when 2/lambda delta x=1; or when delta x~ lambda/2;
So even if you used/collected every propagating plane wave, best possible resolution would be ~l/2.

11. How does diffraction influence image resolution?
Think back to our thought-experiment:
What is the impact of using a lens of diameter D, a distance d away from the screen?
The maximum value of k// is then .
Ask them where k// max comes from
So then sharpest edge will be then roughly

12. Repeat slightly more rigorously for circular distribution of plane waves focussed from a collimated beam through lens diameter D
Recognize this from when we looked at the beam waist of the focussed Gaussian (lecture #14 Additional notes)? There for a Gaussian, estimated 3/pi lambda f/D based on a crude approx.

13. Polarization
Should really spend 3 lectures on this topic, but prefer to use last 3 lectures to do a mini design project.
Proper treatment can be developed using Stokes vectors or Jones vectors, or the Poincare sphere, or …

14. Basics For a plane wave propagating in the positive z direction:
Jones vector

15. Special plane wave polarizations
Linear: Ax and Ay =?
Circular: Ax and Ay =? Ax Ay t
If Ax and Ay in phase (both the same phase factors), they produce linearly polarized light and the relative magnitudes of Ax ad Ay determine what? Ans: the orientation of the polarization.
Before showing 2 and 3, ask what defines a circularly polarized wave?
Right or Left as show in blue? (Ans: left)

16. Generalize
Simple way to generalize is to use plane wave expansion, and let each individual plane wave component have some specified plane wave polarization Jones vector.
Example: Gaussian (0,0) mode with each paraxial plane wave polarized in the plane defined by and : so polarization vector of each paraxial plane wave is to a good approximation: