Download Review of electromagnetic fields

1. REVIEW OF ELECTROMAGNETIC FIELDS
1.1 Maxwell’s Equations
 Review the features of Maxwell’s Equations in differential forms.
 Discuss these equations in different representations: space-time (r, t), spacefrequency (r, ω), wavevector-frequency (k, ω).
1
1.1.1 Maxwell’s Eqs. in the space-time representation
 The first equation, Faraday’s induction law, establishes that a varying magnetic
field produces a rotating electric field,
  E(r, t )  
dB(r, t )
.
dt
(1.1)
 E(r, t) is the electric field (V/m) and B(r,t) the magnetic induction or magnetic
flux density (Wb/m2).
 Ampere’s circuital law states a magnetic field can be generated by existing
currents and by varying electric fields,
  H(r, t ) 
dD(r, t )
 J S (r, t )  J C (r, t ).
dt
(1.2)
 H(r,t) is the magnetic field intensity (A/m), D(r,t) the electric induction or flux
density (C/m2), Js(r,t) the source current density (A/m2), and Jc(r,t) the
conduction current density (A/m2).
2
 Gauss’s law for the electric and magnetic fields:
  D(r, t )   (r, t )
(1.3)
 ρ(r, t) is the volume charge density (C/m3), and
  B(r, t )  0 ,
(1.4)
 which establishes the non-existence of magnetic charge.
 In addition, the following constitutive relations apply:
D   E;    0   r
B   H;   0  r
(1.5)
JC   E
 ε the dielectric permittivity, μ the magnetic permeability, and  the conductivity
 Generally, ε, μ, and σ, are inhomogeneous (r-dependent) and anisotropic
(direction-dependent, tensors). They can also be field-dependent for nonlinear
media.
3
 For linear, homogeneous, and isotropic media, ε, μ, and σ are simple scalars.
The relative permittivity r and permeability μr are defined with respect to the
vacuum values, ε0=10-9/36π F/m and 0  4 107 N/A2.
4
1.1.2 Boundary conditions
 To solve Maxwell’s equations for fields propagating between two media,
boundary conditions for both the electric and magnetic fields are necessary.
Medium 1
E1  E1n  E1t ; D1  D1n  D1t
(s , J s )
n̂
H1  H1n  H1t ; B1  B1n  B1t
E2  E 2 n  E 2t ; D2  D2 n  D2t
Medium 2
H 2  H 2 n  H 2t ; B 2  B 2 n  B 2t
Figure 1. Fields at the boundary between two media
5
 4 equations describe the relationship between the tangent components of E and
H and the normal components of D and B in the two media (Fig. 1), as follows:
nˆ  (E1  E 2 )  0
nˆ  (H1  H 2 )  jS
(1.6)
nˆ  (D1  D2 )   S
nˆ  (B1  B 2 )  0
 n is the unit vector normal to the interface, Js is the surface current, and ρs is the
surface charge density. In the absence of free charge and currents, the boundary
conditions simply imply that the tangential fields (E, H) and normal inductions
(D, B) are conserved across an interface.
6
1.1.3 Maxwell’s equations in the space-frequency representation (r, )
 For broad-band fields, the spatial behavior of each temporal frequency is of
interest. To obtain Maxwell’s equation in the space-frequency representation,
we use the differentiation property of Fourier transforms,
d
 F(t )  i  F( )
dt
(1.7)
 F is the Fourier transform operator (in time), F(t) is an arbitrary time-dependent
vector, and F(ω) its Fourier transform.
 We will denote the Fourier transform of a function by the same symbol, with the
understanding that F(t) and F(ω) are two distinct functions.
7
 Taking the Fourier transform of Eqs. 1-4 and using the differentiation property
in Eq. 7, we can rewrite the Maxwell’s equations in the (r, ω) representation,
  E(r,  )  iB(r,  )
  H (r,  )  i D  J S (r,  )  J C (r,  )
  B(r,  )  0
  D(r,  )   (r,  )
 The constitutive relations take the form
D(r,  )   E(r,  );    0   r ; D   0E  P
B(r,  )   H(r,  );   0  r ; B  0 H  M (1.9)
J C (r,  )    E(r,  )
8
(1.8)
 Even for homogeneous media ε is a function of frequency, which establishes the
dispersion relation associated with the medium. The boundary conditions apply
for each individual frequency,
nˆ   E1 (r,  )  E 2 (r,  )   0
nˆ   H1 (r,  )  H 2 (r,  )   J S
(1.10)
nˆ  (B1  B 2 )  0
nˆ  (D1  D2 )   s
9
1.1.4 The Helmholtz equation
 Eliminating H, B, and D from Eqs. 1-5, we obtain an equation in E(r, t), the
wave equation. The Helmholtz equation can be obtained in the (r, ω)
representation, as follows.
 First let us consider linear, isotropic, and charge-free media (=0). Eqs. 8a-d
simplify to
  E(r,  )  i H(r,  )
  H(r,  )  (  i )E(r,  )
D  0
B  0
(1.11)
 Applying the curl operator to Eq. 11a, we obtain
 E  i  H
(1.12)
10
 Using the identity    E  (E)   2E and the fact that E  0 for chargefree media, we obtain the Helmholtz equation:
 2 E(r,  )   ( ) 2 E(r,  )  0
(1.13)
 ( )    ( )  i ( )
2
2
 In Eq. 13a, the Laplace operator applied to vector E is defined as
 2E   2 Ex  xˆ   2 E y  yˆ   2 Ez  zˆ .
 Equation 13b establishes the frequency dependence of the wavenumber (ω),
the dispersion relation of the medium. Typical biological structures are
characterized by constant μ, the permittivity ε and conductivity σ can have
strong dependence on frequency.
11
 In Cartesian coordinates, the vector Eq. 13a becomes three scalar equations.
 2 Ex (r,  )   2 Ex (r,  )  0
 2 E y (r,  )   2 E y (r,  )  0
(1.14)
 2 Ez (r,  )   2 Ez (r,  )  0
 All the equations are of the same form, referred to as the scalar wave equation,
2U (r,  )   2U (r,  )  0 .
 Assuming solutions of the form U (r,  )  X ( x,  )  Y ( y,  )  Z ( z,  ) , we obtain
three 1D equations, which eventually lead to the plane wave solution of the form
U (r,  )  A  eikr
(1.15)
k x  k y  k z       i 
2
2
2
2
2
12
 Decomposing the wavevector into its real and imaginary parts,
k ( )  k '( )  ik ''( ) ,
(1.16)
 We can rewrite Eq. 15a as
U (r,  )  A  e k ''( )r  eik '( )r
(1.17)
 In Eq. 17 the real and imaginary parts of k capture the refraction (phase term)
and absorption/gain (amplitude term) of the medium.
 In lossless media, k       0 0   r r 
at optical frequencies r
k ( )  n( )
 k0 

c

c

c
  r r . In dielectric media,
1 and  r equals the refractive index, thus
 n( )k0
(1.18)
is the wavenumber in vacuum.
13
1.1.5 Maxwell’s equations in the (k, ω) representation
 Often we deal with optical fields with broad angular spectra. These can be
decomposed in wavevectors k of different directions. The modulus of each kvector is defined by the dispersion relation, depending on the material properties
(ε, μ, σ) and optical frequency ω.
 The normal representation of the fields is in the k-ω space. A differentiation
property similar to Eq. 7 holds for the  operator,
  F(r )  ik  F(k )
  F(r )  ik  F(k )
(1.19)
 The (k, ω) representation of Maxwell’s Equations is obtained by the spatial
Fourier transformation of Eqs. 11a-d.
14
 For media of no free charge (ρ=0) or currents (J=0), these equations are
k  E(k ,  )  B(k ,  )
k  H(k ,  )   D(k ,  )
k D  0
k B  0
(1.20)
 Equations 20a-d describe the propagation of frequency component ω and plane
wave of wavevector k. Eqs. 20c-d establish that k  B and k  D . Generally μ
is a scalar, i.e. B||H, but ε is a tensor, i.e. D is not necessarily parallel to E.
 Thus we see that D  H , such that
k  D&k  H &k  D
(1.21)
 Equations 20a-c show that H, D, and k are mutually orthogonal vectors.
15
H
k
D
Figure 1-2. Mutually orthogonal set of vectors
 For isotropic media, D||E, such that H, E, and k are also mutually orthogonal.
 The characteristic impedance  of the medium is defined as


(1.22)
k
16
 Using Eq. 15b to express k for media with non-zero conductivity, we obtain
1/2


  


  i 


(1.23)
 For non-conducting media (electric insulators), 

. For isotropic media, 

connects the moduli of E and H directly (from Eq. 19b),
E   H
(1.24)
17
1.1.6 Phase, group, and energy velocity
1.0
vg
Re[E(t)]
0.5
0
-0.5
-1.0
-30
vp
-15
0
15
30
t
Figure 3. Phase and group velocity.
 Consider the electric field associated with a light beam propagating along the +z
direction with an average wavevector <k>.
18
 As shown in Fig. 3, the temporal signal has a slow modulation (envelope) due to
the superposition of different frequencies, and a fast sinusoidal modulation
(carrier), at the average frequency <ω>,
E ( z, t )  A( z, t )  ei (  t k  z )
(1.25)
 Thus the phase delay of the field is
 ( z, t )   t  k z
(1.26)
 The phase velocity is associated with the advancement of wave fronts, for which
=constant. Differentiating Eq. 26, we obtain
d ( z, t )   dt  k dz
(1.27)
0
19
 Thus, the phase velocity is

dz
v 

dt
k

(1.28)
c
n
 Equation 28 represents the zeroth order approximation in the expansion:
d (k ) d  c 

  k 
dk
dk  n 
c ck dn
  dn 
  2
 v 1  

n n dk
n
d



 vg
20
(1.29)
 Equation 29 defines the group velocity, vg, at which the envelope of the light
signal propagates. The group and phase velocities are equal in non-dispersive
dn
media (
 0 ).
d
 We prove that the group velocity defines the energy velocity of the field. The
energy velocity is the ratio between the Poynting vector S and electromagnetic
volume density U,
1
S
U
S  E H
1
1
U  ED  HB
2
2
ve 
(1.30)
 We now differentiate Eqs. 20a-b, as follows
 k  E  k   E   H   H
 k  H  k   H   E   E
(1.31)
21
 The meaning of δk and δω is that of a spread in k-vectors (directions) and
optical frequency. Dot-multiply Eq. 31a by H and Eq. 31b by E,
 k  (E  H)  k (H   E)   H  H   H   H
(1.32)
 k  (E  H)  k ( H  E)   (E   E)   (E   E)
 Adding the two Eqs. 32a-b, we obtain
1
2
 k  (E  H)   E   E   H  H 
(1.33)
 using the fact that the products  H( H  k  E) and  E(  E  k  H) vanish.
 Finally, using the definition in Eq. 30a, we find
k
ve 

(1.34)
 vg
22
 Equation 34 establishes the result that the electromagnetic energy flows at group
velocity; it is apparent from Eq. 29 that vg can exceed the speed of light in
dn
vacuum c, in special circumstances, where
 0 (anomalous dispersion).
d
 This does not pose a conflict with the postulate of the relativity theory, which
states that the signal velocity, at which information can be transmitted via
electromagnetic fields, is bounded by c.
23
1.1.7 The Fresnel Equations
 Consider light propagation at the interface between two media (Fig. 4). We will
derive the expressions for the field reflection and transmission coefficients.
x
E TE
n2>n1
H TM
kr
kt
t
t
z
i
ki
n1 n2
Figure 4. Two media of refractive index n1 and n2 separated by the x-y plane. The subscripts i, t, r refer to incident,
transmitted, and reflected.
24
 In Figure 4, the x-z plane is the plane of incidence (plane of the paper), i.e. the
plane defined by the incident wavevector and normal ( n̂ ) at the interface.
 The tangential components of the fields and normal components of inductions
are conserved in the absence of surface currents and charges (Eqs. 6a-d),
nˆ  (E1  E2 )  0
nˆ  (H1  H 2 )  0
(1.35)
nˆ  (D1  D2 )  0
nˆ  (B1  B 2 )  0
 For ρ=0 and J=0, Maxwell’s Eqs. in the k-ω representation yield (Eqs. 19)
H
1

E
k E
1

(1.36)
kH
25
 Expanding the cross products in Eqs. 36a-b, the problem breaks into two
independent cases: a) transverse electric (TE) mode, when E is perpendicular to
the plane of incidence (E||y) and b) transverse magnetic (TM) mode, when H||y.
i) TE mode (E||y)
 If E||y, the boundary conditions for the tangent E-field and normal B-field
E yi  E yr  E yt
(1.37)
H zi  H zr  H zt
 Using Eq. 36b to express Eq. 37b in terms of Ey components, we can rewrite the
system of equations as
k xi  E yi  k xr  E yr  k xt  E yt
(1.38)
E yi  E yr  E yt
 Eqs. 38a-b must hold for any incident field Eyi, we obtain the following result
kxi  kxr  kxt
(1.39)
26
 Eq. 39 establishes Snell’s law,
n1  sin i  n1  sin r  n2  sin t
(1.40)
 using that the wavevector k in a medium of refractive index n relates to the
wavevector in vacuum as k=nk0.
 Eq. 40 implies kzi=-kzn.
 To obtain the field reflection coefficient, we use the continuity of tangent H
components,
H xi  H xr  H xt
(1.41)
 which can be expressed in terms of E components (via Eq. 36b),
k zi E yi  k zr E yr  k zt E yt
(1.42)
27
 Finally, combining Eqs. 42 and 37a to solve for the E field transmission and
reflection coefficients, we obtain
rTE 
E yr
tTE 
E yt
E yi
E yi


k zi  k zt
k zi  k zt
(1.43)
2k zt
k zi  k zt
28
ii) TM mode (H||y)
 Using the analog equations to the TE mode (Eqs. 37a-b), the conservation of Hy
components and normal D components, we find that kx=const. for TM as well.
 To obtain the field reflection and transmission coefficients, we use the
conservation of both the tangents fields components, Ex and Hy,
H yi  H yr  H yt
(1.44)
k zi
k zt
k zr

H


H

 H yt
yi
yr
2
2
2
n1
n1
n2
 The
1
n1,2 2
factor occurs due to the 1/ε factor in Eq. 36b (ε=n2). Thus the H field
reflection and transmission coefficients for the TM mode are
29
rTM 
tTM 
H yr
H yi
H yt
H yi
kiz ktz
 2
2
h
h2
 1
kiz ktz

h12 h2 2
(1.45)
2ktz
kiz ktz
 2
2
h1 h2

 We expressed rTM and tTM in terms of H fields to emphasize the symmetry with
respect to the TE case. Of course, the quantities can be further expressed in
terms of E fields via H 
k E

. Conservation of energy is satisfied in both
cases,
kTE  tTE  1
2
kTM  tTM
2
2
2
(1.46)
1
30
 Together, Eqs. 43a-b and 45a-b, the Fresnel equations, provide the reflected and
transmitted fields for an arbitrary incident field. Because of the polarization
dependence of the reflection and refraction coefficient, polarization properties of
light can be modified via reflection and refraction. In the following we discuss
two particular cases that follow from the Fresnel equations, where the
transmission or reflection coefficients vanish.
31
1.1.8 Total internal reflection
 Setting tTE=tTM=0 yields the same condition for “no transmission” in both TE
and TM modes, transmission coefficient vanishes, kzt=0. Thus,
k zt  kt 2  k xt 2
 n2 2  k0 2  n12  k0 2  sin 2 i
(1.47)
0
 using kx=constant (Eq. 38), kxt=kxi. The transmission vanishes for
 n2 
C  sin  
 n1 
1
(1.48)
 θc is the critical angle at which total internal reflection takes place. Total
internal reflection can occur for both TE and TM polarizations, the only
restriction being that n2<n1.
32
 For angles of incidence that are larger than the critical angle, θi>θc, the field
reflection coefficient becomes
k zi  i k zt
rTE 
k zi  i k zt

 TE
(1.49)
 iTE
e
 i 2TE

e
eiTE
 k zt 
 tan 
.
 k zi 
1
 For θi>θc, the reflection coefficient is purely imaginary, the power is 100%
reflected, but the reflected field is shifted in phase by 2TE.
33
 Similarly, for the TM mode we obtain
rTM  e  i 2TM
TM
 k zt  n12
 tan 
 2
k
 zi  n2
(1.50)
1
 Since TM and TE have different values, total internal reflection can be used to
change the polarization state of optical fields.
 The transmitted plane wave has the form
Et  E0  eik zt z
 E0  e
(1.51)
 k zt  z
 Equation 51 indicates that the field in medium 2 is decaying exponentially.
34
Ezt
z
1/kzt
Figure 1-5. Evanescent field decaying exponentially with depth z.
 Thus the field is significantly attenuated over a distance on the order of 1/kzt, i.e.
the field does not propagate, or is evanescent.
35
1.1.9 Transmission at Brewster angle
 Another case of Fresnel’s equations is when the reflection coefficient vanishes.
 For the TE mode, we have
rTE  0  kzi  kzt  n1  n2
(1.52)
 For TE polarization, the only way to obtain maximum transmission through an
interface is when there is no refractive index contrast between the two media,
the trivial solution.
 For the TM mode the situation is very different:
rTM  0 
kiz ktz
 2  n2  cosi  n1  cost
2
h1
h2
(1.53)
 The condition is satisfied simultaneously with Snell’s law, such that we have
n1 sin i  n2 sin t
(1.54)
n2 cosi  n1 cost
36
 Multiplying Eqs. 54a and 54b side by side, we obtain
sin 2i  sin 2t  i  t 

(1.55)
2
 The angle of incidence at which rTM=0, referred to as the Brewster angle, is
defined by combining Eq. 55 and Eq. 54a
tan  B 
n2
n1
(1.56)
 Unlike with total internal reflection, where the transmission can vanish for both
polarizations, the reflection can only vanish in the TM mode.
37
 The absence of reflection at the Brewster angle for TM polarization can be
understood by the absence of radiation by an (induced) dipole along its axis
(Fig. 6). The concept of induced dipoles followed by re-radiation is essential for
the Lorentz model of light-matter interaction, as detailed in the next section.
x
kt
z
H
ki
Figure 1-6. Brewster angle incidence.
38
1.2 The Lorentz Model of Light Matter Interaction
 Review the main concepts in basic atom-field interactions. In particular the
Lorentz model, a pre-quantum mechanics model, and its asymptotic case for
metals, the Drude model.
 The Lorentz model explains much of classical optics via a physical picture
borrowed from mechanics. The starting point is the “mass on a spring”
description of electrons connected to nuclei. Thus, the incident electric field
induces displacement to the electron that is under the influence of a spring-like
restoring force due to the nucleus.
39
 The equation of motion for the electron can be expressed as
d 2 x(t )
dx(t )
e
2




x

E (t ),
0
2
dt
dt
m
(1.57)
  is the damping constant, 0 is the resonant frequency, e is the electronic
charge, m mass of the electron, and E the incident field.  and 0 are
characteristics of the material, the first describing the energy dissipation
property of the medium and the second the ability of the medium to store
energy.
 Since Eq. 57 is a linear differential equation, Fourier transforming both sides of
the equation gives the frequency-domain solution.
40
 Using the Fourier property of the differential operator,
dn
n

(
i

)
,
n
dt
(1.58)
 we obtain Eq. 57 in the frequency domain
e
 x( )  i x( )  0 x( )   E ( ) .
m
2
2
(1.59)
 Thus, we find the solution for the charge displacement in the frequency domain,
e
E ( )
x( )  2 m
.
2
  i  0
(1.60)
 To obtain the time domain solution, x(t), we need to Fourier-transform Eq. 60.
However, we explore further the frequency domain solution. The induced dipole
moment due to the charge displacement x() is
p( )  e  x( )
(1.61)
41
 In Eqs. 59-60 we obtained microscopic quantities, the atomic-level response.
 The macroscopic behavior of the medium is obtained from the induced
polarization P, which captures the contribution of all dipole moments within a
certain volume,
PN p .
(1.62)
 N is the volume concentration of dipoles (m-3) and the angular brackets denote
ensemble average.
 Assuming that all induced dipoles are parallel within the volume, we obtain
Ne2
E ( )
P( ) 
 2
.
2
m 0  i  
(1.63)
42
 Generally, each atom has multiple resonances or dipole-active modes, such that
Eq. 63 can be generalized to
 E ( )
Ne2
P( ) 
 2 i 2
,
m i 0i    i i
(1.64)
 The summation is over all modes, characterized by different resonant
frequencies and damping constants. The weight i is called the oscillator
strength and has the quantum mechanical meaning of a transition strength.
 For simplicity, we reverse to the single normal mode description, which captures
the origin of absorption and refraction of materials. The induced polarization
only captures the contribution of the medium itself, it excludes the vacuum
contribution.
43
 Thus,
P  0E
(1.65)
   r  1  n  1.
2
  is the dielectric susceptibility, which generally is a tensor quantity.
 However, for isotropic media, we obtain the complex scalar permittivity
 r   r '( )  i r ''( ,
Ne2
1
 r ( )  1 
 2
m 0 0   2  i
(1.66)
Ne 2

has units of frequency squared,  p 2 , and p is the plasma frequency.
m 0
44
 From Eq. 66, we readily obtain the real and imaginary parts of r,
0 2   2
 i '( )  1   p
(o 2   2 )   2 2

 r '( )   p 2
(0 2   2 ) 2   2 2
2
(1.67)
 Figure 7 illustrates the main features of  ' and  '' vs. frequency. To gain further
physical insight into Eqs. 67a-b, we discuss three different frequency regions, as
follows.
45
1.2.1 Below the resonance,   0
 In this case, Eqs. 67a-b simplify to
p2
1
 r '( ) 1  2 
0 1  2  / 0 2
(1.68)
 p  
1
 r ''( )

0 4 1  2( / 0 ) 2
2
 Since   0 2 ,  '   '' , absorption is negligible, below the resonance the
material is transparent. Further, d  '
d
dispersion.
46
 0 , which defines a region of normal
 It can be seen that expanding the denominator in Eqs. 68a-b, we obtain
 r '( )   2 and  r ''( )   3 . In designing optics of imaging systems, the
Sellmeier equation is very efficient for describing the refractive index vs.
wavelength,
2
a

n 2 ( ) 1   2i
i   bi
 1 
i
,
ai
1  bi 2 /  2 c 
(1.68a)
2
 The summation is over several resonances, ai and bi are experimentally
determined parameters, and c is the speed of light. It can be seen that the
Sellmeier equation originates in the expression for  r '( ) in Eq. 1.68.
 As we approach resonance, this dependence becomes more complicated.
47
1.2.2 At resonance,  0
 For frequencies comparable to 0 , Eqs. 67a-b are well approximated by
 p2
 r '( ) 1 

20
p

20
2
 r ''( )
  0
   0 
1 


/
2


2
(1.69)
 /2
   0 
1 


/
2


2
 Under these conditions, the absorption is significant,   0   , and the
absorption line has a characteristic shape, Lorentzian line. This shape has a
central frequency 0 and a full width half maximum of  .
48
 While 0 has a clear physical significance of the frequency at which the system
“resonates”, or absorbs strongly, the meaning of  is somewhat more subtle.
 The damping constant  represents the average frequency at which electrons
collide with atoms, which generates loss of energy. Thus,   1/  col , with  col
the average time between collisions.
 Finally, around resonance, d  '
d
 0 , which defines anomalous dispersion.
49
1.2.3 Above the resonance,   0
 Well above the resonance, the following equations apply:
2
 r '( ) 1   p  4
   2 2

 r ''( ) 1   p 2  4
   2 2
2
(1.70)
 The absorption becomes less significant, as expected in a frequency range away
from the resonance. The dispersion is normal again, d  '
d
 0.
 This Lorentz oscillatory model provides great insight into the classical lightmatter interaction. In the following section, we will investigate the particular
situation of metals, when the charge moves freely within the material.
50
1.3 Drude model of light-metals interaction
 The optical properties of metals were first introduced by Drude in the context of
conductivity. In highly conductive materials, the restoring force in Eq. 57,
m0 2 x , vanishes, establishing that the charge can move freely. Under these
conditions, we obtain Drude’s model, in which Eqs. 67a-b reduce to (0  0)
 p2
 r '( )  1  2
  2
(1.71)
p

 2
   2
2
 r ''( ) 
 Typically   1/  col   , the frequency of collisions is much lower than that
of optical frequencies.
51
 In this high frequency limit,  r '( ) 1 
p2
2
and  r ''( )
 p 2
3
. From the
Fresnel equations, we derive the reflectivity coefficient. For normal incidence,
the intensity-based reflectivity is
n 1
R( ) 
,
n 1
2
(1.72)
 n is the (complex) refractive index. Since n   r , Eq. 72 becomes
R ( ) 
 r ' i r ''  1
 r ' i r ''  1
2
(1.73)
52
 Figure 7 illustrates the frequency dependence of  ' ,  '' , n’, n’’ and R for various
values of  p and  .
r’’
r’
R()
p
Figure 7. Frequency dependence of dielectric permeability and reflectivity around the plasma frequency (p =10, =1) .
53
 At    p ,  r '( ) vanishes. In this case, the real part of the refractive index, n’,
can also vanish. This implies that the wavelength in the material is infinite,
  0 / n   .
 To gain a physical understanding of the plasma frequency  p , consider a thin
film of metal.
x{ + + + + + + + + + + + + + + + +
E
P
- - - - - - - - - - - - - - - -
Figure 1-9. Exiting surface plasmon resonance.
 The applied electric field induces a polarization P   0  E , with    r  1.
54
 Tuning the frequency of the incident field to the plasma frequency,  r  0 ,
  1, and P   0E . The induced polarization is the total charge times the
displacement per unit volume,
P   N  e x
(1.74)
 Therefore, the electric field is
E

P
0
Ne
0
(1.75)
 x
 If we construct the electric force due to the charge displacement, F  eE , we
obtain
F 
Ne2
 x
0
 ke   x.
(1.76)
55
 In Eq. 76, we define ke as the “spring” constant of the restoring force. By
definition, the system is characterized by a resonant frequency,  p  ke / m .
This is the plasma frequency associated with the thin film,
Ne2
.
p 
m 0
(1.77)
 From Maxwell’s equations, we have that   H 
D
E D
. At plasma
 0

t
t t
frequency, D   0 E , and the magnetic field vanishes. This indicates that there
is no bulk propagation of electromagnetic field.
 Since the E-field must be in the plane of the surface (plane of incidence), it is
clear that only TM polarization can induce surface plasma resonance. The
quantum of energy associated with the charge oscillations at plasma frequency is
 p and the respective quantum particle is called plasmon.
56
 Light scattering on plasma oscillations, including in nanoparticles (Figure 9) can
be described as a photon-plasmon interaction. These interactions are the basis
for many modern sensors and lab-on-a-chip devices.
+ + + +
E
P
- - - -
Figure 1-10. Plasmon resonance in nanoparticles.
57