1

Chapter 3 & 4Beam Optics + Fourier Optics

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 kx

2+ky2+kz

2=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 Phasekx

2 + ky2 + kz

2 = k2

OL 26, 485 (2001)

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

Expansion based on the solutions of wave equation -- plane waves

Paraxial Optics? 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?

x

kx

Sharp edge high spatial frequency

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

Perfect lense / superlensePhys. Rev. Lett. 85, 3966–3969 (2000), cited by 4500 times

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