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```Chapter 3 & 4
Beam Optics + Fourier Optics
1







3.1 THE GAUSSIAN BEAM
Under paraxial Helmholtz equation
Properties
W0 ~ z01/2

Gouy Phase
Divergence and Gouy Phase

Dispersion Relation
kx2+ky2+kz2=k02
 Isotropic media

Divergence and Gouy Phase
 Confinement in x-y
 w(z)
 1/w(z)
 Divergence and Gouy
Phase
Gouy Phase
OL 26, 485 (2001)
kx2 + ky2 + kz2 = k2

Problems



Could explain the divergence!
Wrong for Gouy phase


Weights of different Fourier components do not vary
with distance z!
Why?

Paraxial Approximation
A dilemma exists! 也许值得探讨
3.2 TRANSMISSION THROUGH
OPTICAL COMPONENTS

3.4 LAGUERRE-GAUSSIAN AND
BESSEL BEAMS

Vortex
Generation
Applications
Bessel Beams
< k, Gouy Phase
Non-integrable
FOURIER OPTICS

based on


harmonic analysis (the Fourier transform)
linear systems
Expansion Methods

Fourier Optics


Paraxial Optics?



Expansion based on the solutions of wave equation
-- plane waves
LG waves
HG waves
Expansion based on the orthogonal and
complete sets
4.1 PROPAGATION OF LIGHT IN
FREE SPACE







Fourier Transformation中, k=n2π/A, 取分立值

4.2 OPTICAL FOURIER TRANSFORM
4.3 DIFFRACTION OF LIGHT
WHY?
Sharp edge  high
spatial frequency
x
kx
Problem?

We must utilize the components with higher
wavevectors kx, ky>k

kz become pure imaginary!

Evanescent waves & surface waves
Bigger k  smaller wavelength
 电镜, x-ray
SNOM
Phys. Rev. Lett. 85, 3966–3969 (2000), cited by 4500 times
Perfect lense / superlense

negative refractive media）材料可以实现超聚焦
science_308_534 (2005)
END
Homework

Plot the curves of Eqs. (3.1-8) to (3.1-10)
versus z, and explain what they means

EXERCISE 3.1-3

EXERCISE 3.1-2
Homework


EXERCISE 4.1-1
What is Fresnel Approximation and Fraunhofer
approximation? Explain their difference

EXERCISE 4.2-2

EXERCISE 4.3-3
```
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