2. Types of field distribution:
Gaussian: can solve directly the paraxial (Fresnel) approximation)
Beams and pulses – most often treated in Gaussian beams/pulses – space-time analogy applies.
Simpler approximation: Fraunhofer. The FAR FIELD is the Fourier transform of the “object”
It applies to amplitude and phase object. The “object” has 3 dimensions: x, y, and phase.
We are still dealing with coherent light.

3. One approach to find the reflectivity/transmission of a very thin film is to
use the Fabry-Perot transmission function in the limit thickness d -> 0.
Thin dielectric layer
d and d  0:
Thin gain layer

4. Non-reciprocity: a dielectric film followed by a gain film:
one can have zero reflection from the left, finite reflection from the right,
identical transmission from both sides.
= 0 if
Example:

5. Macroscopic structure with gain, reflection from on side, none from the other side:
replace the gain layer by multiple gain layers separated by a half wavelength,
and the dielectric layer by multiple layers separated by a half wavelength.
Put in a ring cavity: unidirectional oparation.

7. Fraunhofer diffraction to Fourier transform to imaging

8. Fraunhofer and Fourier – what is the connection?
Given: Field in plane z=0
e(x,y)
Find: Field in plane z=L
e(x’,y’,z) y y’
kAP = kR - kyy q
AP = R – y sin q P A R q z O L x’ x

10. e(x,y) Case 3: lens f kx = (k/f)x’ ky = (k/f)y’ P k y f ky A
Recipe: take the Fourier transform, and replace v u

11. Random distribution of identical objects
Types of field distribution:
Gaussian: can solve directly the paraxial (Fresnel) approximation)
Beams and pulses – most often treated in Gaussian beams/pulses – space-time analogy applies.
Simpler approximation: Fraunhofer. The FAR FIELD is the Fourier transform of the “object”
It applies to amplitude and phase object. The “object” has 3 dimensions: x, y, and phase.
We are still dealing with coherent light.
Random distribution of identical objects
Resolution and transfer function
Elementary image: a grating.

13. Example of application: the GLORY
Far field distribution of a random ensemble of identical objects

14. H. C. Bryant and N. Jarmie, “The {Glory}",
Scientific American, 231:60—71 (1974)
Other application: monitoring the growth of
water drops in clouds (more spaced resonances)
Monitoring the evaporation of a large droplet
The successive narrow peaks are separated by 1/100 wavelengths,
indicating that they correspond to resonances of 100 cycles.
Backscattered
intensity
TIME

15. How to monitor the size of a water droplet
as it growth or evaporates?
Through whispering gallery modes

16. Water droplet suspended from a glass thread returns a beam lf laser light straight back to the observer.
Droplet is 2 mm DIA; thread is 40 mm thick.
1.153 mm DIA
741 mm
20 minutes

17. Fraunhofer diffraction

18. The Glory
A complete description of the glory involves not only the whispering gallery resonance,
but also the fact that the backscattering is issued from multiple light »rings«
(the annular edge of the droplets), which interfere with each other.
Simulation:
Make negative
Sum of rings different size?
FT of randomly located
Small objects?
Effect of colors?
Effect of phases?

21. Not to be confused with the rainbow

22. Rainbow Angle always the same (except in LA) Incoherent sum
Broad distribution of angles
up to a sharp maximum
The concept of “maximum deviation”
Independent of the size
of the droplets

23. Double rainbow

24. And not to be confused with the Halo
Related to the minimum deviation of water crystals

27. Glory Diffraction from rings
Rainbow
The concept of “maximum deviation”
Halo
The concept of “minimum deviation”