Ray Theory of Waves

Ray Theory of Waves

Kuan Fang REN

CORIA/UMR 6614 CNRS - Université et INSA de Rouen

School of physics and optoelectronic Eng., Xidian University

I. Fundamentals of geometrical optics and physical optics

Lecture at Xidian University

on Frontiers in modern optics

波的射线理论

任宽芳

第一讲：几何光学和物理光学基础

西安电子科技大学现代光学前沿专题

法国鲁昂大学 — CORIA研究所

西安电子科技大学物理与光电学院

个人网页：http://ren.perso.neuf.fr

项目网页：http://www.amocops.eu

电话：中国 8820 2673，法国 02 32 95 37 43

http://www.amocops.eu/

西安电子科技大学现代光学前沿专题Lecture at Xidian University

Plan of lecture

I-3

➢ Introduction

▪ What’s the light?

▪ Reflection, refraction, diffraction and scattering

➢ Geometrical optics

▪ Properties of light rays

▪ Snell’s laws

▪ Fresnel formulas

➢ Physical optics

▪ Huygens-Fresnel principle

▪ Phase of ray and interference

➢ Summary


Introduction

I-4

➢ 有了光才有了宇宙，才有了人类，• 盘古开天后我们才有了光，有了天地，有了阴阳

• 上帝说要有光，就有了光，然后才有亚当和夏娃

➢ 光是认识和改造世界的工具• 从日常生活、光通讯、天文学…到核能、暗物质

光是什么？


Introduction

I-5

➢ 光的射线模型—几何光学（古老而唯像的描述）

➢ 光仅是电磁波中很窄的一个波段—波动理论(经典物理观) 可见光波长：400-800纳米，约头发丝直径的1/100

➢ 光是人类思想发展的动力▪ 19世纪末物理学的两朵乌云催生了两门新学科：

迈克耳孙 → 莫雷实验光速不变 → 相对论

黑体辐射 → 紫外灾变 → 量子力学

▪ 相对论的时空观，广义相对论，宇宙起源

▪ 量子理论，光子纠缠、量子通讯、信息的超距传输

➢ 对光的本质的认识还在继续：暗物质，暗能量 … …

光是什么？


Introduction

I-6

光的反射和折射➢ 均匀介质中直线传播

➢ 遇到界面发生折射

眼睛通过光学系统，如镜子，眼镜，看物体时，看到的是它的像，像的位置取决于进入眼睛的光线。

镜子的最小尺寸与距离有关吗？

路面的光反射

•氦氖激光在水中的波长是多少？•在水中看是什么颜色？


Introduction

➢ 光照通过一个狭缝或遇到遇到细丝会发生明显的衍射• 可用于测量微小物体的尺寸（头发丝）：d = D l / D

光的衍射和散射

➢ 圆孔衍射和瑞利判据

▪ 两光斑的最小分别角


GO: Properties of light rays

➢ Four properties of a geometrical ray:

• Direction: Reflection a refraction

• Amplitude: Reflection a refraction

I-8

Rainbow:- Direction- Amplitude- Phase- Polarization- Curvature of

wavefront

Rainbow:- Direction- Amplitude- Phase- polarization

What difference between the near field and the far field?

• Phase

• Polarization

• Curvature of wavefront


GO: Properties of a light ray

➢ Four properties of a geometrical ray:

• Eikonal equation

• (𝛁𝐿)𝟐= 𝑛𝟐 or 𝛁𝐿 = 𝑛ො𝒔

where ො𝒔 =𝒅𝒓

𝒅𝑠=

𝛁𝐿

𝛁𝐿is the normal of the wave front.

𝐿 = 𝑛𝒌 ∙ 𝒓 is called eikonal→ optical path.

In a homogeneous medium, the ray propagates straightly.

• Tube of rays

R are the curvature radii of the wave fronts.𝐼(𝐵)

𝐼(𝐴)=

𝐺,𝐵

𝐺,𝐴

𝐺 =1

𝑅1𝑅2is the Gaussian curvature of the wave fronts.

I-9

葛德彪、魏兵《电磁波理论》第九章Born & Wolf, Principles of Optics, sect. 3.1

ො𝒔

𝒌

Light ray

𝒓

O

Derive these equations yourself.


GO: Snell’s laws

➢ Snell’s laws

The plane of incidence is defined by incident ray and the normal of the diopter.

I. The reflection law:1. The reflected ray is in the plane of incidence.2. The reflection angle is equal to the incident angle

𝜃𝑖 = 𝜃𝑙II. The refraction law:

1. The refracted ray is in the plane of incidence.2. The refraction angle 𝜃𝑖 is related to the incident angle 𝜃𝑟 by

𝑚0 sin 𝜃𝑖 = 𝑚𝑝 sin 𝜃𝑟

where 𝑚0 and 𝑚𝑝 are the refractive indices in

the incident and refracted media.

I-10

DemonstrationPhase match: mpOE=m0BC

DemonstrationPhase match: OD=BC


GO: Snell’s laws

➢ Formation of image (plane surface)

• Perpendicular position𝜃𝑖 → 0𝑑 = 𝐻𝜃𝑖 = 𝐻′𝜃𝑖𝑚𝜃𝑖 = 𝑚′ 𝜃𝑟

So 𝐻

𝑚=

𝐻′

𝑚′

• Arbitrary positionGeometry :

𝑑 cos 𝜃𝑟 = 𝑂𝐴′𝑑𝜃𝑟 , 𝑑 cos 𝜃𝑖 = 𝑂𝐴𝑑𝜃𝑖Refraction law:𝑚1 cos 𝜃𝑖 𝑑𝜃𝑖 = 𝑚2 cos 𝜃𝑟 𝑑𝜃𝑟

Eample: 𝜃𝑖=45°, 𝑚

2

𝑚1

= 1.333, 𝜃𝑟 = 32°:

𝑂𝐴′= 1.58 𝑂𝐴

I-11

𝑚1cos2𝜃𝑖𝑂𝐴′ = 𝑚2cos

2𝜃𝑟𝑂𝐴

d

ImageImage

H H’

m’

m


GO: Snell’s laws

➢ Formation of image (Curved diopter)

• Paraxial case

» Spherical diopter

(cf. figure on the bottom)

» Thin lens equation

• Off-axis case

― Horizontal position?

― Vertical position

I-12

B

F O

F’

A

A’

B’ J

I

'

11

'

1

fOAOA=−


GO: Fresnel formulas

➢ Fresnel formulas (transparent media)

I. Perpendicular polarization:

I-13

to derive the equations by yourself.




I. Parallel polarization:

I-14

to derive the equations by yourself.




Perpendicular case:

I-15

Attention to the reference direction of the field E.




Reflectivity and transmissivity:Poynting vector:

Energy flux per unit area:

So the reflectivity and transmissivity:

I-16

Attention: T t2

but T =(1‒r2)




Reflectivity :

I-17

Brewster angle

Air to water Water to airlimit angle

➔ Calculate the three angles.

Forbidden zonen’>n

n

n

n’<n




Amplitude and phase of rays :

I-18

Attention: When the incident angle is bigger than the critical angle, the phase varies continuously.


PO: Huygens-Fresnel principle

➢ Diffraction

• Diffraction is the behavior of waves when they encounter an obstacle.

• The effect of diffraction is significant when

» the size of the object is of the order of the wavelength

» the change in density/amplitude is abrupt.

I-19

➔ Calculate the three angles.

A wave encounters an obstacle. The effet of diffraction in rainbow.Diffraction by a double slit.



➢ Principle of Huygens-Fresnel

1) The contribution of Huygens (1678)

The wave of light propagates step. Each surface element reached by it behaves as a secondary source that emits spherical wavelets. The new wavefront is the envelop of the wavelets and the amplitude of each wavelet is proportional to the size of the element.

2) The contribution of Fresnel (1818)The complex amplitude of the wave at one point is the

sum of the complex amplitudes of all the secondary

sources at that point. All these waves interfere to form

the wave at the considered point.

I-20

t+Dt

t



➢ Theory of diffraction

• Fresnel diffraction (R is small):

• Fraunhofer diffraction (R infinity):

I-21



➢ Applications

• Diffraction by circular disk (sphere):

A circular hole of radius r :

Because of the revolution symmetry, we can place in the plane of the disk with the origin at the center:

I-22

cos , sin ,x y dS dxdy d d = = = =

( )2

1

0

sin

sin

JI I

=

=2pr/l

𝑎 = sin , β = 0

Rayleigh criterion:

The first minimum of J1(x) is

located at x=3.832, i.e.:

1.22

D

l =



➢ Applications

• Diffraction by a single slit:

• Yong’s slits: N=2

• N slits:

I-23

2

0 2

sin uI I

u=

Width of the central spot: 0

2DL

a

l=

fringe :D

ib

l=

22

0 2

sincos

u bI I u

au

=

2 2

0 2 2

sin sin

sin

u NvI I

u v=

sin sin,

a bu v

p p

l l= =


PO: Phase of ray and interference

➢ Time factor – time convention: 𝑒−𝑖𝜔𝑡 or 𝑒+𝑖𝜔𝑡

• The sign of the phase depends on the time factor.

• The direction of propagation also.This wave propagates in +z axis: 𝐸 = 𝐸0𝑒

𝑖(𝑘𝑧−𝜔𝑡), 𝐸 = 𝐸0𝑒𝑖(𝜔𝑡−𝑘𝑧)

While this wave propagates in +z axis: 𝐸 = 𝐸0𝑒𝑖(𝑘𝑧+𝜔𝑡), 𝐸 = 𝐸0𝑒

−𝑖(𝜔𝑡+𝑘𝑧)

➢ Phase due to reflection• Phase of p: The time factor has no importance.

it seems there is no “half wavelength loss” in English or French.

• But the sign of the phase shift in total reflection.

➢ Phase of optical path• The supplementary path causes a positive phase shift.

• The wave needs more time to arrive at the same position.

I-24

z𝐸1 = 𝐸0𝑒

𝑖(𝑘𝑧1−𝜔𝑡)

𝐸1

𝐸2

z1

𝐸2 = 𝐸0𝑒𝑖(𝑘𝑧

1+D−𝜔𝑡)


Summary

➢ Geometrical optics

• Properties of rays:

» direction, amplitude, phase, polarization

• Snell’s law:

» Reflection law, refraction law (attentions : the incident plane).

• Fresnel formulas:

» The polarization is important

» Special angles and change of phase

» The reflectivity and the transmissivity and their variation (T t2)

➢ Physical optics

• Principle of Huygens-Fresnel

» Condition of application

» Approximation methods

» Typical results and their physical significant.

I-25

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Ray Theory of Waves

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