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 W 0 ~ z 0 1/2 束缚越强，扩散越大！

Gouy Phase

Divergence and Gouy Phase  Dispersion Relation  k x 2 +k y 2 +k z 2 =k 0 2  Isotropic media

Divergence and Gouy Phase  Confinement in x-y  w(z)  Broadening in k x -k y  1/w(z)  Divergence and Gouy Phase

Gouy Phase k x 2 + k y 2 + k z 2 = k 2 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 / superlense Phys. 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  EXERCISE 3.1-2

Homework  EXERCISE  What is Fresnel Approximation and Fraunhofer approximation? Explain their difference  EXERCISE  EXERCISE 4.3-3