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Transcript 
Chapter 3 & 4
Beam Optics + Fourier Optics
1
Comments

第二章的延续

非平面波


主要沿z方向传播
在横截面（xy）里，电磁场为非均匀分布
3.1 THE GAUSSIAN BEAM
Under paraxial Helmholtz equation
Properties
W0 ~ z01/2
束缚越强，扩散越大！
Gouy Phase
Divergence and Gouy Phase

Dispersion Relation
kx2+ky2+kz2=k02
 Isotropic media

Divergence and Gouy Phase
 Confinement in x-y
 w(z)
 Broadening in kx-ky
 1/w(z)
 Divergence and Gouy
Phase
Gouy Phase
OL 26, 485 (2001)
kx2 + ky2 + kz2 = k2
源于量子受限效应
Problems

Broadening in wavevector?


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
光路设计上
很重要
3.4 LAGUERRE-GAUSSIAN AND
BESSEL BEAMS
伴随着l，存在环状能流，
在光学蜗旋上、光镊里起
到关键作用！
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
算法实现

周期边界性条件



取样区域的尺寸大小A为周期
Fourier Transformation中, k=n2π/A, 取分立值
可调用Fast Fourier Transformation (FFT)命
令
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
用左手材料/负折射率（left-handed materials or
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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