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.

SPACE TIME Fourier transform in time Fourier transform in space

TIME SPACE

From Huygens Integral to Fresnel integral: In one dimension: From Fresnel integral Integral to Fraunhofer: Fresnel integral changed into a Fourier transform: Fraunhofer:

Fraunhofer diffraction to Fourier transform to imaging

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

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

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.

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

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

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?

Resolution and transfer function More general than just imaging. Time equivalent: Signal I(t), analyzed with an instrument that has not sufficient resolution – gives S(t) Can you recover the original signal? Only if you know the transfer function. The transfer function H(t) is the delta function response of the instrument I(t) Instrument S(t) Instrument H(t)

Point Spread Function Object = convolution of the field with a delta function. Image of the delta function = point spread function H(u,v) Image of the object = | Fourier transform of object x H(u.v) |2