Lecture 02 : Chapter1. The propagation of light

EE 430.423.001 2016. 2^nd Semester

2016. 9. 6.

Changhee Lee

School of Electrical and Computer Engineering Seoul National Univ.

chlee7@snu.ac.kr

Lecture 02 : Chapter1. The propagation of light

EE 430.423.001 2016. 2^nd Semester

Maxwell equations for the vacuum

0 0











 H E





• Gauss' laws

• Faraday's law: electromagnetic induction

t E

H



 







 

 _μ

_{ 4}  _{ 10}

_{H/m  1 . 26  10}

_H/m

• Ampere-Maxwell equation: Maxwell added the time rate of change of electric field (displacement current)

t H

E



 





 



At a point in empty space, the EM state of the vacuum is specified by two vectors, the electric field E and the magnetic field H.

t J E

B



 







 

   

F/m 10

854 .

8 

o





Permeability of the vacuum

Permittivity of the vacuum

EE 430.423.001 2016. 2^nd Semester

Maxwell’s wave equations

2 2 2

2 2

2 2

2

Laplacian

) ( )

(

) (

z y

x 

 



 



 























• From the Maxwell equations, we can get EM wave equations

• Maxwell's discovery merged the fields of electricity, magnetism, and optics.

2 2 2 2

2

2

1

, )

( t

E E c

t E

E



 

 

 











 

 





2 2 2 2

2

2

1

, )

( t

H H c

t H

H



 

 

 











 

 





s m c

o o

/ 10

1 3





  

speed of EM wave (light)

http://micro.magnet.fsu.edu/primer/java/wavebasics/index.html

EE 430.423.001 2016. 2^nd Semester

Measurement of speed of light: Interferometry

A coherent beam of light (e.g. from a laser), with a known

frequency (n), is split to follow two paths and then recombined.

By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light (λ) can be determined. The speed of light is then calculated using the equation c = λ n.

https://en.wikipedia.org/wiki/Speed_of_light

EE 430.423.001 2016. 2^nd Semester

Measurement of speed of light: rotating mirror

Fizeau–Foucault apparatus : On the way from the source to the mirror, the beam passes through a rotating cogwheel. At a certain rate of rotation, the beam passes through one gap on the way out and another on the way back, but at slightly higher or lower rates, the beam strikes a tooth and does not pass through the wheel. Knowing the distance between the wheel and the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light can be calculated.

https://en.wikipedia.org/wiki/Speed_of_light

EE 430.423.001 2016. 2^nd Semester

EE 430.423.001 2016. 2^nd Semester

KK

u n  c 

index of refraction

Speed of light in a medium

o

m

μ

K  μ

o

K 

 

Relative permeability Relative permittivity

speed of light in a medium

K c K

K u K

m o

o m

1 1

1  

   

EE 430.423.001 2016. 2^nd Semester

Dispersion

dispersion : The refractive index of materials varies with the wavelength (and frequency) of light.

This causes prisms and rainbows to divide white light into its constituent spectral colors.

https://en.wikipedia.org/wiki/Refractive_index

chromatic aberration: Dispersion also causes the focal length of lenses to be wavelength dependent.

normal dispersion: The refractive index decreases with increasing wavelength.

EE 430.423.001 2016. 2^nd Semester

https://en.wikipedia.org/wiki/Speed_of_light red

blue yellow

n n

V n



  1

Abbe number

Abbe number

In optics and lens design, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion (variation of refractive index versus wavelength), with high values of V indicating low dispersion.

Most of the human eye's wavelength sensitivity curve, shown here, is bracketted by the Abbe number reference wavelengths of 486.1 nm (blue) and 656.3 nm (red)

EE 430.423.001 2016. 2^nd Semester

Plane harmonic waves.

Waves in 1-dim.

avenumber) (angular w

2

, frequency) (angular

2

velocity) (phase

), cos(

) , (





n

















k u k

t kz

U t

z

U

2 2 2 2

2

1

t U u

z U



 





T u

uT k

1 2

2













 n 

 

t u z  



EE 430.423.001 2016. 2^nd Semester

Waves in 3-dim.

z y

x o

k k

k k

z y

x r

t r

k U

t z y x U

kˆ jˆ

iˆ

, kˆ jˆ

iˆ

) cos(

) , , , (



















 

 



Plane harmonic waves.

Surfaces of constant phase

constant













 r t k x k y k z t

k ^{ } 



2 2

2

z y

x

k k

k k

u      

EE 430.423.001 2016. 2^nd Semester

Sources of EM waves

EE 430.423.001 2016. 2^nd Semester

Alternative ways of representing harmonic waves







cos i sin

e

 

• Traveling wave in complex notation

)

)

, ( ) cos(

) ,

( z t U

kz t U z t U

e

U     

)

)

, , ,

( x y z t U

e

U 

• Spherical wave

)

1

),

1 cos(

r e t

r kz





EE 430.423.001 2016. 2^nd Semester

Group velocity

• Superposition of two waves

] ) (

) [(

] ) (

)

)

,

( z t U

e

U

e

U 



• Group velocity

dk d u

 k 

 

 

) cos(

2 ] [

) ,

( z t  U e

e

 e

 U e

z  k  t   U

• phase velocity

u _  k

 

 

  





 



 



 

 dk

dn n u k

dk dn n

ck n

c n

kc dk

d dk

u

d 

1

o o

g

g

d

dn c

u u

d u du

u 



   



 1 1

,

• For most optical media the index of refraction increases with increasing frequency (normal dispersion), so that dn/dk >0 and therefore u_g<u.

(Prob. 1.6)

EE 430.423.001 2016. 2^nd Semester

Doppler effect

 

 



     

 



 





  1

' c

u c u u

c

c n

n n

 

 

  

 



 



  

c u c

u

c 1

' n n

n

If the source is moving away from the receiver with a velocity u,

If the receiver is moving away from the source with a velocity u,

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/dopp.html

c

 u

 



n n n

n n '

other each

departing when

'

other;

each g

approachin when

'

 

 







 

 

 







 

source receiver

source receiver

u c

u c

u c

u c

n n

n

n

EE 430.423.001 2016. 2^nd Semester

Doppler shifts of spectral lines in astronomy

https://ase.tufts.edu/cosmos/print_images.asp?id=44

EE 430.423.001 2016. 2^nd Semester

Relativity correction

From the Einstein's Theory of Special Relativity, the clock on a moving source runs slow by a factor of g=(1-u²/c²)^-1/2.

Thus, the rate at which new wave crests appear - the frequency - is slowed down by g=(1-u²/c²)^-1/2.

c u

c c u

c u u c

u c

u u

c c

/ 1

/ / 1

/ 1 1

1 1

/ 1

' 1 /

1

' 1



 

 



 





 

 

 





 

 



 





 

 

 





  n n

n g n n

n n

 

 



     

 

 

2 1 1

/ 1

/ ' 1

c u c

u c

u c

u n

n n

other) each

departing when

: other;

each g

approachin when

: (

' -

u c

u c  

 n 

n

EE 430.423.001 2016. 2^nd Semester

Experimental verification of relativistic Doppler effect

 

 



     

 

 

2 1 1

/ 1

/ ' 1

c u c

u c

u c

u n

n n

H. E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 31, 369 (1941)

 

 



  



 

 



  



2 2 2 2

2 1 1

2 1 1

c u c

u

c u c

u

reflect direct









Relativistic correction

EE 430.423.001 2016. 2^nd Semester

Doppler broadening of spectrum lines

m kT c

2 ln 2

 2

 n

n

http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT404/HTML/AT4040 4.htm

Molecular speed distribution

dv e

kT v N m

dv v

n ( )  4 ( )

 

EE 430.423.001 2016. 2^nd Semester

Homework set #1.

Solve Problems 1.2, 1.3, 1.4, 1.6, 1.7, 1.10, 1.11.