ELECTROMAGNETIC WAVES IN MEDIUM
•	The electric susceptibilities are in general  tensors
•	P = ε0 χe E
•	M = χm H
•	For isotropic media the magnetic and electric susceptibilities reduce 
•	from a tensor to a scalar

MEs In a Linear, Isotropic Medium

•	( ε0 E) = ‑ (χeε0 E) + ρ 
•	( μ0 H )= ‑ ( χmμ0 H)
•	x(E) = ‑  ( μ0 H ) ‑  ( χmμ0 H )
•	x (H) =  ( ε0 E ) + ( χeε0 E ) + σE
•	This might have some solutions!!

MEs In a Linear, Isotropic Medium

•	(E) = ρ/ε 
•	Div( H ) = 0
•	x ( E ) = ‑ ( μ H )
•	x (H) =  ( ε E )  + σE
•	 ε = ε0 ( 1 + χe )
•	 μ = μ0 ( 1 + χm )

•	For a linear, isotropic, charge‑free medium 

•	2 (H) ‑ M2/Mt2 ( μεH ) ‑ M/Mt ( μσH ) = 0
•	2 (E) ‑ M2/Mt2 ( μεE ) ‑ M/Mt ( μσE ) = 0

•	The third term in the above equation
      represents the dissipation
•	For a linear, isotropic, charge‑free, 
     non‑conducting medium ( σ = 0 ) the wave 
     does not dissipate
•	A Vacuum is a particular case of the above!!
•	
Transverse Waves

•	2 ( H ) ‑ M2/Mt2 ( μεH )  = 0
•	2 ( E ) ‑ M2/Mt2 ( μεE ) = 0

     The most general plane wave solutions of the 
     above are
•	E = x [ f+ (z ‑ vt) + f‑ (z + vt) ]

•	H = ( ε/μ )1/2 y [ f+ (z ‑ vt) + f‑ (z+vt) ]

•	Note that E and H are perpendicular to 
      each other and to the direction of 
      propagation ‑ z
•	
Monochromatic Waves

•	  E = E0 x cos( k .  r  " ωt + φ)
•	 φ is an arbitrary phase
•	 ω is the angular frequency
•	 k defines both the direction of propagation
     (DOP) and the wave vector |k| = 2π/λ = k
•	k . r " ωt defines a surface of constant 
     phase which is displaced in time at a 
     speed ω/k which is called the Aphase speed@

Velocity In a Medium

•	If χe, χm are # 0, then the phase speed is 
      reduced
•	Phase speed becomes c/n
•	n = ( ε/ε0 . μ/μ0 )2
•	n is the Arefractive index@ and is > 1 (mostly)
•	If either of χe, χm are dependent upon
     frequency, then the medium has Adispersion@
•	This means that if n is frequency‑dependent, 
     then the medium is dispersive
•	
Complex Numbers

•	Complex numbers for physicists are a 
     Aconvenience@ for solving problems
•	Rely on the real‑world fiction that %‑1 can be represented by i (the engineers use j)
•	A complex number can be written as
     C = a + ib
•	The conjugate complex is C* = a ‑ ib
•	The modulus is |C| = %(a2 + b2 )
•	The argument is Arg(C) = tan‑1 (b/a)
•	
Complex Exponentials - Phasors

•	 If C = cos(a) + i sin(a)
•	Then it is shown in all the best text books that
     you can write C = exp(ia)
•	In fact any complex number C can be
     written as C = |C| exp(i arg(C))
•	Since manipulation of exponentials in 
      integration and differentiation is very easy, it 
      is tempting to try to manipulate any wave
      equation into this format 
•	
Complex Superposition

•	If   E0 x cos (k . r ‑ ωt + φ ) is a solution
•	Then so is E0 x sin (k . r ‑ ωt + φ ) 
•	And so is   i . E0 x sin (k . r ‑ ωt + φ )
•	By the principle of linear superposition, 
     the sum of the above solutions is also 
     a solution which can be expressed as
     E0 x exp i(k . r ‑ ωt + φ )

Differentiation wrt Time

•	For any quantity C exp(‑iωt) where C is 
     independent of time
•	M/Mt (C exp(‑iωt) ) =   ‑iω  C exp(‑iωt) 
•	Similarly Int[C exp(‑iωt) ] = ‑1/iω C exp(‑iωt)
•	What does this do to Maxwell=s equations? 
  MEs
•	( ε0 E) = ‑ P + ρ  - unchanged
•	( μ0 H) ] = ‑ [ μ0 M ) - unchanged
•	x E = ‑ M/Mt ( μ0 H ) ‑ M/Mt ( μ0 M ) becomes
•	x E = iωμ0 H + iωμ0 M
•	x H =  M/Mt ( ε0 E ) + M/Mt ( P ) + J becomes
•	xH = - iωε0 E - iω P + J 
•	
ME Monochromatic Wave Eqn

•	2 ( H ) + ω2μεH + iωμσH = 0
•	2( E ) + ω2μεE + iωμσE = 0
•	μ0 = 4π . 10‑7 in SI units
•	ε0 = 10‑9/(36π) in SI units
•	If we now REDEFINE ε as being ε + iσ/ω, we 
     can automatically incorporate conducting
     media in the solutions with a complex 
     dielectric constant
•	This implies that the effect of conductance
       is the the polarisation P is still proportional 
      to the field E, but there is a phase lag between
      the field and the polarisation
•	 We can also look at the same arguments for
       the   spatial differential 2 and find that in a 
       charge‑free medium 2 [X] = -k2 X where
       k is the wave vector
•	
Dispersion Relation

•	Putting all these components into the
      wave equation, we find that
•	k2 = ω2 ε μ for a linear, isotropic, charge‑free
     medium
•	Remember that ε is generally complex 
     and therefore so is k
•	If we define refractive index to be complex following ε, then

      k2 = ω2 /c2 .  n2

•	This is the Adispersion relationship@ and 
      measures how the wave vector k, varies
      with  frequency
•	
n and ε

•	Most materials are non‑magnetic, assume μ=μ0
•	ε = εr + i εi  = n2 ε0
•	So if n = nr + i ni
•	Then   εr / ε0 = nr2 ‑ ni2 
•	And    εi / ε0 = 2 nr ni 
•	What=s the point?
•	We can measure the refractive index very
     easily
•	therefore we can relate the macroscopic n 
     to the microscopic, complex ε
•	
Bringing It Back to Reality

•	The wave propagates like exp i( k .  r ‑ ωt + φ)
•	Suppose it propagates along the z axis -> kz
•	The wavelength in free space is λv 
•	and the wavevector is kv
•	Look at the spatial part only
•	exp(ikz)  = exp{i ( nr + i ni ) kv z }
•	exp(ikz) = exp (i nr kv z)   exp(‑  ni kv z)
•	First term is phase factor, second is a decay 
     term
•	Measure the phase speed ‑ gives nr 
     (historically n)
•	Measure the decay ‑ gives ni (historically κ)
•	
Refractive Indexes

•	Vacuum = 1.00 
•	Glasses = 1.57‑1.77
•	Sapphire = 1.77, Diamond = 2.42
•	Silicon = 3.8 + 0.4i
•	Silver = 2.3 + 3.8i
•	Air = 1.0 + 290 .  10‑6 ( ρ/ρstp)