Introduction to Photonics lecture 13-14-15 16 Electromagnetic Optics(1)

8/10/2019 Introduction to Photonics lecture 13-14-15 16 Electromagnetic Optics(1)

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Introduction to Photonics

Lecture 13/14/15/16: Electromagnetic OpticsNovember 3/5/10/12, 2014

• Electromagnetic theory

• Review of Maxwell’s equations

• Electromagnetic waves in dielectric media

• Conductive media

• Time-harmonic ME and TEM waves

• Absorption and dispersion

• Resonant media and Lorentz model 1



2

Electromagnetic Optics

• Ray optics! light as ray theory – Reflection, refraction, imaging

– Good approximation when wavelength is small in comparison to size of

optical component

• Wave optics! light as scalar wave theory – Ray optics plus diffraction and interference (by considering phase)

• Electromagnetic optics! light as vector wave theory

– Wave optics plus fraction of light reflected and/or transmitted (by

considering polarization)



• Light is an electromagnetic phenomenon.

• An electromagnetic field is described by two mutually coupled

vector fields, the electric field, E , and the magnetic field, H .

– E = ( E x , E y , E z) and H = ( H x , H y , H z) – Wave optics is a scalar approximation of electromagnetic optics.

• Relationship between electric and magnetic field vectors

described by Maxwell’s equations

3

Electromagnetic Optics



4

Review of Maxwell’s Equations

• E (r, t ) , H (r, t ) are electric and magnetic fields – Units of E : [V/m]; Units of H : [A/m]

– Six functions of E and H must satisfy M.E.

• D(r, t ) is electric flux density (or displacement

field) [C/m2]

• B(r, t ) is magnetic flux density [W/m2 = T]

• ! is electric permittivity [F/m]

– ! 0 = 8.854 x 10-12 F/m

• µ is magnetic permeability [H/m]

– µ 0 = 4" x 10-7 H/m

• J (r, t ) is current density [A/m2]

• # (r, t ) is charge density [C/m2]

• P(r, t ) is polarization density [C/m2]

• M (r, t ) is magnetization density [A/m]

• Many variables, however, this is for any arbitrary medium

• Relationships simplify for many media

0),(),(),(

),(),(

),(),(

),(

=$%=

$%

&

&'=(%

+&

&=(%

t Bt t D

t

t Bt E

t J t

t Dt H

rrr

rr

rr

r

#

),(),(),(

),(),(),(

t M t H t B

t Pt E t D

rrr

rrr

µ µ

!

+=

+=



6

E&M Waves in Dielectric Media

• The constitutive relations describe the specific medium; theydescribe how currents and charges are generated

• Have relation between P and E and between M and H for medium

),(),(),( t Pt E t D rrr += ! ),(),(),( t M t H t B rrr µ µ +=

Describes dielectric properties Describes magnetic properties

),(),(

1

),( t M t Bt H rrr '

= µ

H is medium independent

From Maxwell’s equations

D is medium independent

Constitutive Relations



Nondispersive:

Material is memoryless, i.e., P at time t is related

to E at same time, but not to prior values

Linear

Nonlinear1.

Linear:

Any component of P is weighted superposition

of components of E

Nondispersive

Dispersive

4.

Isotropic

Anisotropic3.

Isotropic:System is invariant to rotation of coordinate

system. P is parallel to E

Homogeneous

2.

Homogenous:

System is invariant to displacement; relationbetween P and E is independent of position

Nonhomogeneous

7

Properties of Dielectric Media



8

Maxwell’s Equations

0),(

),(),(

),(),(

),(),(

),(

=$%

=$%

&

&'=(%

+&

&=(%

t B

t t D

t t Bt E

t J t

t Dt H

r

rr

rr

rr

r

# Divergenceequations

Curlequations

Time-varying displacement (electric

field) and current are accompanied by

rotating magnetic field

Time-varying magnetic fieldaccompanied by rotating electric field

Diverging displacement (electric field)

related to charge (both bound and free)

Diverging magnetic flux densitydoesn’t exist



9

Review of Differential Operators

k z

j y

i x &

&+&

&+&

&)%del

Gradient

(slope) k

z j

yi

x &

&+

&

&+

&

&)%

* * * * * is scalar field

Divergence

(degree of outwardness) z

V

y

V

x

V V

&

&+

&

&+

&

&)$%

321 k V j V iV V 321 ++= is a vector field

Uses of del:

0=$% V 0+$% V



10


k z

j y

i x &

&+&

&+&

&)%del

Curl(degree of rotation) ( )

k y

V

x

V j

x

V

z

V i

z

V

y

V

V V V

z y x

k j i

k V j V iV k z j yi xV

!! "

#$$%

&

&

&'

&

&+!

"

#$%

&

&

&'

&

&+!!

"

#$$%

&

&

&'

&

&=

&

&

&

&

&

&

=++(

!! "

#

$$%

&

&

&

+&

&

+&

&)(%

123123

321

321

Uses of del:

0+(% V 0=(% V



11


Laplacian

(divergence of

the gradient)!!

"

#$$%

&

&

&+!!

"

#$$%

&

&

&+!!

"

#$$%

&

&

&=%)%$%

2

2

2

2

2

22)(

z y x

* * * * *

2

2

2

2

2

22

z y x &

&+

&

&+

&

&)%

Important identities

V V V

V 2)()(

0)(

0)(

%'$%%=(%(%

=(%$%

=%(% *



12

M.E. 1: Ampere’s Law

Displacement current(added by Maxwell)

),(),(

),( t J t

t Dt H r

rr +

&

&=(%

Conductioncurrent

• A time-varying displacement (electric

field) OR current generates magnetic

field with rotation

• Another way to state ‘generates’ is to say ‘is

accompanied by’

• The displacement current term allows forexistence of electromagnetic waves (when

J = 0, E still accompanied by H )

• Recall curl is a measure of field rotation –

how much vector curls around a given point



13

M.E. 2: Faraday’s Law

t

t Bt E

&

&'=(%

),(),( r

r

• A time-varying magnetic flux (magnetic field) generates an

electric field with rotation• Rotating current (field) on a solenoid generates magnetic field

• Time-varying magnetic field generates counter current due to

rotating (curl) electric field



14

M.E. 3: Gauss’s Law

• The divergence of the electric displacement (electric flux) isproportional to the charge density present at the point

• Mathematical expression for Coulomb’s law

• Relates charge to electric field

• Recall that divergence is a measure of total outgoing flux per unit

volume

),(),( t t D rr # =$%



15

M.E. 4: Absence of Magnetic Charges

• There are no points in space that act as sources/sinks formagnetic field lines (or magnetic flux)

• Magnetic field lines always close on themselves

• There are no magnetic charges (magnetic monopoles)

0),( =$% t B r



Introduction to Photonics

Lecture 13/14/15/16: Electromagnetic OpticsNovember 3/5/10/12, 2014

• Electromagnetic theory

• Review of Maxwell’s equations

• Electromagnetic waves in dielectric media

• Conductive media

• Time-harmonic ME and TEM waves

• Absorption and dispersion

• Resonant media and Lorentz model 16



17

Review of Maxwell’s Equations

• E (r, t ) , H (r, t ) are electric and magnetic fields – Units of E : [V/m]; Units of H : [A/m]

– Six functions of E and H must satisfy M.E.

• D(r, t ) is electric flux density (or displacement

field) [C/m2]

• B(r, t ) is magnetic flux density [W/m2 = T]

• ! is electric permittivity [F/m]

– ! 0 = 8.854 x 10-12 F/m

• µ is magnetic permeability [H/m]

– µ 0 = 4" x 10-7 H/m

• J (r, t ) is current density [A/m2]

• # (r, t ) is charge denstiy [C/m2]

• P(r, t ) is polarization density [C/m2]

• M (r, t ) is magnetization density [A/m]

• Many variables, however, this is for any arbitrary medium

• Relationships simplify for many media

0),(

),(),(

),(),(

),(),(

),(

=$%

=$%

&

&'=(%

+&

&=(%

t B

t t D

t

t Bt E

t J t

t Dt H

r

rr

rr

rr

r

#

),(),(),(

),(),(),(

t M t H t B

t Pt E t D

rrr

rrr

µ µ

!

+=

+=



Nondispersive:

Material is memoryless, i.e., P at time t is related

to E at same time, but not to prior values

Linear

Nonlinear1.

Linear:

Any component of P is weighted superposition

of components of E

Nondispersive

Dispersive

4.

Isotropic

Anisotropic3.

Isotropic:System is invariant to rotation of coordinate

system. P is parallel to E

Homogeneous

2.

Homogenous:

System is invariant to displacement; relationbetween P and E is independent of position

Nonhomogeneous

18

Properties of Dielectric Media



Linear, Nondispersive, Homogeneous,

Isotropic Media

%2u - (1/c2)&2u/ &t 2=0 c = (! µ )-1/2 = co /n

, !

! +=== 1

0

0

c

cn

0µ µ =non-magnetic materials:

0

0

0

=$%

=$%

&

&'=(%

&

&=(%

H

E

t

H E

t

E H

µ

!

)1(0

0

, ! !

!

!

+=

=

=

E D

E P

Contains only

E, H fields



Time-Harmonic Fields

A monochromatic (time-harmonic) field can be written as:

Spatial complex amplitude

])()([2

1})(Re{),(

t j t j t j eeet E - - - '.+== rErErEr

)cos()(

)sin()}(Im{)cos()}(Re{),(

/ -

- -

+=

+=

t

t t t E

rE

rErEr

This is a different way of writing:

where !! "

#$$%

& = '

)}(Re{

)}(Im{tan( 1

rE

rEr)/



M.E. for Time-Harmonic Fields

Expressed with only complex spatial amplitudes

where complex permittivity is:

0 (1 ) i0

! ! , -

= + +

B

!D

BE

JDH

$%

=$%

'=(%

+=(%

-

-

j

j

MHB

PED

00

0

µ µ

!

+=

+=



Plane Harmonic Waves

Plane harmonic (monochromatic waves) are defined as:

M.E. for plane monochromatic waves are derived using properties of differential operators:

constant vector

!differential operators act as a multiplication operation

From M.E. we see that:

0

0

0 0

0 0

0

0

k E

k H

k E H

k H E

- µ

-!

$ =

$ =

( =

( = '

)(

0),( rk t j

e E t E $'= -

r r jk e E

$'= 0)(rE

complex amplitude

jk

j t

1%

1&

&-



Helmholtz Equations

For source-free media, and time-harmonic fields:

Typical scattering problem

0

0

22

22

=+%

=+%

HH

EE

k

k

Vector Helmholtz equations:

! µ - 22 =k

What are the boundary conditions ?

%$%)%2 is a dyadic which, when operated

on a vector yields a vector.

In the special case where E is specified relative to a rectangular

Cartesian coordinate system ! the components of E separatelysatisfy the scalar wave equation:

022 =+% 2 2 k



0=$%

=$%

B

D

0

3

=$

=$

''

'''''

&

&

V

V V

dAn B

rd dAn D #

( )( ) 012

12

=$'

=$'

n B Bn D D 0

t

B E

t D J H

&

&'=(%

&&+=(%

'''

'''

$! "

#$%

&

&

&'=$

$! "

#$%

&

&

&+=$

&

&

S S

S S

dAnt

Bdl E

dAnt

D J dl H

( )( ) K H H n

E E n

='(

='(

12

12 0

Using the divergence theorem

:K Surface

current density :0 Surface charge

density

Using Stokes’s theorem

Boundary Conditions



Boundary Conditions

From M.E. 1, 2:

From M.E. 3, 4:

( )( )

0i j

i j

n E E

n H H K

( ' =

( ' =

( )( ) 0

i j

i j

n D D

n B B

0 $ ' =

$ ' =

a surface current

density may be present

a surface charge

density may be present

Tangential components

Condition:

Normal components

Condition:



Tangential components of E, H" continuous

Normal components of D, B" continuous

Boundary Conditions



The simplest solution of Maxwell’s equations in free space is the harmonic transverse

electromagnetic wave (TEM) wave

TEM Wave

3 o = µ o ! o = 377 Ohms

Properties:

• E and H are orthogonal to one another and to the direction of propagation (the z direction)

• The ratio of the magnitudes of E and H is the impedance = 377 Ohms.

z

x

y

E

H

H

z

E

E = E oe j - (t ' z/c o ) ˆ x

H = E o3 o

e j - ( t ' z/ co ) ˆ y

3 o = µ o ! o



Poynting Vector and Energy Flow

S E H = ( Poynting vector (power flux density, W/cm2)

[ ]1

2W D E B H = $ + $ E&M energy density (J/cm3)

Specifies the transfer rate of E&M energy: of fundamental importance in propagation, absorption, scattering

The rate at which E&M energy is transferred across a general surface:

A

W S ndA= ' $'

Positive if energy is absorbed inside V

For harmonic fields in linear media, the time-averaged Poynting vector is:

{ }1

Re ( ) *( )2

S E r H r= (This is a relevant experimental quantity:=),( t r I

Optical intensity



Intensity Relations for TEM Waves

.(= H E S

2

1

3 2

2

0 E I = 2

02

1 E W ! =

cW I =Time-averaged power density flow =

transport of time-averaged energy density

at the velocity of light



= 4 + j 4 4

k = - ! µ o = 1+ , k o = 1+ 4 , + j 4 4 , k o

k = 5 ' j 12 6 5 ' j 12 6 = k o 1+ 4 , + j 4 4 ,

n ' j 6

2k o= 1+ 4 , + j 4 4 ,

Absorption is governed by a complex susceptibility:

Complex wavenumber

Let

6 = absorption coefficient , attenuation coefficient, or extinction coefficient

U = e' jkz

= e' j ( 5 ' 1

26 ) z

= e' 1

26 z

e' j 5 z I = U

2= e

'6 z

5 = propagation constant = nk o

Expression relating refractive index and

absorption coefficient to complex

susceptibility

z

Attenuated wave

Absorption

then,

Expression for plane wave:

! = ! o(1 + , )Complex permittivity



Strongly Absorbing Media

'1'' , , +>>

2/)''(2

2/)''(

0 , 6

,

'8

'8

k

n

00'' >(< 6 Note that

)''()1(2

1)''(''2/ 0 , , , 6 ''±=''=8' j j j k j n



Remarks on Conventions

In general:

n

5

6 are REAL quantities

3

!

k are COMPLEX quantities

'''1/2

10

0

, , ! ! 6

j k

j n ++=='6 5 2

1 j k '=

,

3

!

µ 3

+

==

1

00Complex impedance:0nk = 5

Propagation constant

(effective refractive index):

Complex wavenumber:



Complex Refractive Index

With 0 ( ) E E f k r t - = $ ' we can obtain the dispersion relation

2 2

2 ( )

c k ! -

- =

complex electric permitivity

( ) ( )n n i- 9 ! - = + =!Where

Complex refractive index

)(~)( - - !

-

n

cc

k ==

(relation between k and - )



Complex Refractive Index

( ) ( )n n i- 9 ! - = + =!

0

( )n

: :

- =

0

4( ) ( )

" 6 - 9 -

:

=

Snell’s law refractive index

(real dispersion)

Intensity extinction

(energy dissipation)

Re( ) ( ) ph

g

cv

k n

vk

-

-

-

= =

&=

&

Absorption coefficient

)(~)( - - !

- n

cck

==



Alternative Relations

2 ( )n ! - =! ( )

( ) ' ''

n n i

i

- 9

! - ! !

= +

= +

!

2 2'

'' 2

n

n

! 9

! 9

= '

=

( ) ( )

( ) ( )

2 2

2 2

' '' '

2

' '' '

2

n! ! !

! ! ! 9

+ +=

+ '=

Or, equivalently:

( , ) ( ', '')n 9 ! ! ;

connection describes the intrinsic

optical properties of matter

! +=1

Complex refractive index



Kramers-Kronig Relations

Hilbert transform pairs

For any linear, shift-invariant, causal system with real impulse-response functions,

, ’ and , ’’ are related through Kramers-Kronig relations:

Kramers-Kronig relations for n and 6

Relation between absorption and dispersion: if n(: )!

then 6 (: )



Optical Transmission



P

Origin of Frequency Dependence of Susceptibility

Regard the P-E relation

as a (dynamic) linear system

!Since , is frequency dependent,

n and 6 are also frequency dependent.n(- ) ' j 6 (- )

2- co

= 1 + 4 , (- )+ j 4 4 , (- )

P (t ) =! o ' , (t ' 4 t )E ( 4 t )d 4 t

Dielectric mediumLinear, homogeneous, isotropic

})(Re{)( t j et E - - E= )()()( 0 - -- - - -- - ! !! ! - -- - EP =



LSI Systems and Dielectric Media

Electric dipole

)(t P : response (real)

)(t E : input (real)

)(0 t ! : impulse-response (real)

)(0 < ! : transfer function

*)()( < , < , =' : Hermitian symmetry

P ( t ) =! o ' , (t ' 4 t )E ( 4 t )d 4 t

)(t 7

Recall LSI = linear, shift-invariant

P P = - Nex

N : atomic density

-e: electronic charge

x: charge displacement



Resonant Medium: Lorentz Model

NpP

ex p

=

'= induced dipole moment/atom

(-e = electron charge; x = displacement)

polarization density

( N = atomic density)

Lorentz model of materials

K

,m e“There is not a single granule oflight which is not the fruit of an

oscillating charge.” - A. Lorentz

How do we model the dipole moment (of an atom) induced by incident electric field?

! model motion of bound charge as driven harmonic oscillator

t j

cloc e E E

-

= : local electric field that induces dipole moment



Lorentz model of materials

K

,m e

With harmonic driving field:

2

0 /

/

K m

b m

-

=

=

=

driving force

Solution:


loceE Kxdt

dxb

dt

xd m '=++

2

2

damping constant

resonant frequency

loceE xmdt

dxm

dt

xd m '=++ 2

02

2

- =

t j cloc e E E - = t j

c e x x - =

)( 22

0 =- - - j m

eE x c

c+'

'=Note: complex representation of real

time-harmonic quantities



43

loceE xm

dt

dxm

dt

xd m '=++ 2

02

2

- =


mass*

acceleration

Where does this come from?

restore frictionexternaltotal

total

F F F F

maF

++=

= Newton’s Law

Driving force from

electric field of

incident wave

Friction/damping

(loss: due to

emission,

scattering/

collisions)

Restoring

force



Collection of Oscillators

Induced dipole moment of an oscillator is:

Polarization density of medium containing N oscillators/unit volume is:

Where we have defined the plasma frequency:

22

0

p

Ne

m-

! =

0P E ! , =

ex p '=

Nex NpP '==

E j

P p

022

0

2

! =- - -

- +'

=

Recall:=- - -

- ,

j

p

+'=

22

0

2



Dielectric Permittivity of Oscillators

2

2 2

0

1 1 p

i

- ! , - - =-

= + = +' '

' ''i! ! ! = +

( )( )

( )

2 2 2

0

22 2 2 2

0

2

22 2 2 2

0

' 1

''

p

p

- - - !

- - = -

- =- !

- - = -

'= +

' +

=

' +

1

0P E ! , =



The Resonant Medium

&2 x

&t 2 +0

& x

&t +- o

2 x=

e

mE

, (< ) = , o< o

2

< o2

' < 2+ j < ><

P

&2P

&t 2 +0

&P

&t +- o

2P =- o

2! o , oE

, o = e2 N

m! o- o2

Electric Dipole

>< = 0

2" Lorentzian susceptibility

m = mass of bound electron,

9 = elastic constant of restoring force

- o =(9 /m)1/2 = resonance frequency

0 = damping coefficient

Solutions of the form:

{ }t j

t j

et P

et E

-

-

P

E

Re)(

Re)(

=

=

P = Nex

0P E ! , = "

- < 2

00 =



Structure of the Lorentz Susceptibility

<

<

>= 0Q

" <

2=>

Low frequency limit

)/12/(0 QQ "±Max/min of :)(' < , occurring at : Q/110 "<

2

00

2

0

- !

,

m

Ne=

Low-frequency

susceptibility

, (< ) = , o< o

2

< o2

' < 2+ j < ><

Plots of real and imaginary parts

Peak of - , ’’

For < >> < 0, , ’ 8 , ’’ 8 0

4 , (< ) = , o

< o2

< o2

' < 2( )

< o

2' < 2( )

2+ < >< ( )2

4 4 , (< ) = ' , o< o

2< ><

< o2

' < 2( )2

+ < >< ( )2



Dielectric Permittivity Approximations

For frequencies in the vicinity of 0-

( )

( ) ( )

( ) ( )

0 0

2 2

0

0

2 2

0

/ 2' 1

/ 2

/ 4''

/ 2

p

p

- - - - !

- - =

=- - !

- - =

'8 +

' +

8

' +

2

pmax

0

0

max max

min max

''

FWHM( '') / 2

' 1 '' / 2

' 1 '' / 2

- ! =-

! - - =

! !

! !

8

1 ' = ±

= +

= 'Lorentzian lineshape

(



Absorption/Refraction

One resonance

(dilute concentration

of atoms)

Multiple resonances

(different lattice and

electronic vibrations)

! Overall susceptibility

from superposition of

resonances

'''1/2

10

0 , , ! !

6 j k j n ++=='

Recall: , ’’ confined near

resonances; , ’ near and

below resonance

!Absoption anddispersion strong near

resonance;

Away from resonance n

constant (nondispersive,

nonabsorptive)



Transparent [in Visible]

Although medium is approximately nondispersive/nonabsorptive away from resonance,

each resonance contributes to value of refractive index away from [below] resonance

Recall Kramers-Kronig relations:



Absorption/Refraction

One resonance

(dilute concentration

of atoms)

Multiple resonances

(different lattice and

electronic vibrations)

! Overall susceptibility

from superposition of

resonances

'''1/2

10

0 , , ! !

6 j k j n ++=='

Recall: , ’’ confined near

resonances; , ’ near and

below resonance

!Absoption anddispersion strong near

resonance;

Away from resonance n

constant (nondispersive,

nonabsorptive)



The Sellmeier Equation (far from resonance)!susceptibility is a sum of terms

Media with multiple resonances

n2(: ) 8 1 + , i

: 2

: 2

' : i2

i

)

n28 1+ , oi

< i2

< i2

' < 2

i)

• Electronic vibrations• Lattice vibrations

Superpositionof different contributions

The imaginary part is confined near resonance

The real part contributes at all frequencies

The Sellmeier Equation

oi



N = n ' : dn

d : D: = ' :

c

d 2n

d : 2

= group index

c/N = group velocity

= dispersion coefficient(ps/km-nm)

Refractive Index of Silica Glass in the Transparent Region

(well described by three resonances)

, 1=.6961663; : 1 =.0684043 µm

, 2 =0.4079426; : 2 =0.1162414 µm

, 3 =0.89794; : 3 =9.896161 µmn 2(: ) 8 1 + , i

: 2

: 2

' : i2

i)

D :

n

N



Refractive Index Dispersion



58

Optics of Conducting Media

EH eff j -! =(%

0µ ! - eff k =

Describes wave propagation with

eff !

µ 3 0=

n and 6 determined from

00 /2/ ! ! 6 eff k j n ='-

0 ! ! !

j reff += 0

When conductive effects dominate

-

0 !

j eff 8

0 - µ 3

0 - µ 6

-! 0

2/)1(

2

2/

0

0

0

j

n

+8

8

8

, where



Physical Meaning

Where complex permittivity defined as:

bound charge current density

free charge current density

Im( ) Im( ) Re 0

! , -

& #= + $ !

% "

Both conductivity and susceptibility contribute to the imaginary part of the permittivity

If the imaginary parts of µ or ! are nonzero, the amplitude of a plane wave will

decrease as it propagates in the medium! absorption

Complex phenomenological coefficients of a medium are equivalent to a phase

difference between P and E (or H and B) and are manifested by absorption

! "

#$%

& ++=

-

0 , ! ! j )1(0

Introduction to Photonics lecture 13-14-15 16 Electromagnetic Optics(1)

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Transcript of Introduction to Photonics lecture 13-14-15 16 Electromagnetic Optics(1)