Maxwell's Equations and Light Waves Longitudinal vs. transverse waves Div, grad, curl, etc., and the 3D Wave equation Derivation of the wave equation from Maxwell's Equations Why light waves are transverse waves Why we neglect the magnetic field Irradiance, superposition and interference Longitudinal vs. Transverse waves Motion is along the direction of propagation Motion is transverse to the direction of propagation Space has 3 dimensions, of which 2 directions are transverse to the propagation direction, so there are 2 transverse waves in ad- dition to the potential longitudinal one. Transverse: Longitudinal: Vector fields Light is a 3D vector field. A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space. () fr ! ! A light wave has both electric and magnetic 3D vector fields: x y A 2D vector field 2 2 2 2 2 2 2 2 2 2 2 0 0 E E t E E E E x y z t μ! μ! " # $ = " " " " " + + $ = " " " " ! ! ! ! ! ! ! The 3D vector wave equation for the electric field which has the vector field solution: Note the vector symbol over the E. ( ) ( ) , exp Ert A ikr t ! " # $ = % & & ’ ( !" !" "" " # ( ) ( ) 0 , exp Ert E ikr t ! " # = $ % & ’ !" "" " " # # This is really just three independent wave equations, one each for the x-, y-, and z- components of E.
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Maxwell's Equations and Light Waves
Longitudinal vs. transverse waves
Div, grad, curl, etc., and the 3D Wave equation
Derivation of the wave equation from Maxwell's Equations
Why light waves are transverse waves
Why we neglect the magnetic field
Irradiance, superposition and interference
Longitudinal vs. Transverse waves
Motion is along
the direction of
propagation
Motion is transverse
to the direction of
propagation
Space has 3 dimensions, of which 2 directions are transverse to
the propagation direction, so there are 2 transverse waves in ad-
dition to the potential longitudinal one.
Transverse:
Longitudinal:
Vector fields
Light is a 3D vector field.
A 3D vector field
assigns a 3D vector (i.e., an
arrow having both direction
and length) to each point in
3D space.
( )f r! !
A light wave has both electric and magnetic 3D vector fields:
x
y
A 2D vector field
2
2
2
2 2 2 2
2 2 2 2
0
0
EE
t
E E E E
x y z t
µ!
µ!
"# $ =
"
" " " "+ + $ =
" " " "
!! !
! ! ! !
The 3D vector wave equation for the
electric field
which has the vector field solution:
Note the vector symbol
over the E.
( ) ( ), expE r t A i k r t! "# $= % & &' (
!" !" " ""
#
( ) ( )0, expE r t E i k r t!" #= $ %& '
!" " """
# #
This is really just
three independent
wave equations,
one each for the
x-, y-, and z-
components of E.
Waves using complex vector amplitudes
We must now allow the complex field and its amplitude to be
vectors:
( ) ( )0, expE r t E i k r t!" #= $ %& '
! !! !!
" "
0 (Re{ } Im{ }, Re{ } Im{ }, Re{ } Im{ })x x y y z z
E E i E E i E E i E= + + +
!
"
The complex vector amplitude has six numbers that must be
specified to completely determine it!
0E!
E!
Note the arrows
over the E’s!
x-component y-component z-component
Div, Grad, Curl, and all that
Types of 3D vector derivatives:
The Del operator:
The Gradient of a scalar function f :
The gradient points in the direction of steepest ascent.
, ,x y z
! "# # #$ % & '# # #( )
!
, ,f f f
fx y z
! "# # #$ % & '# # #( )
!
If you want to know
more about vector
calculus, read this
book!
Div, Grad, Curl, and all that
The Divergence of a vector function:
yx zff f
fx y z
!! !" # $ + +
! ! !
!!
The Divergence is nonzero
if there are sources or sinks.
A 2D source with a
large divergence:
Note that the x-component of this function changes rapidly in the x
direction, etc., the essence of a large divergence.