Laws of Reflection & Refraction
Reflection law: angle of incidence=angle of reflection
Snell’s law of refraction:
n1 sin 1  n2 sin 2
[18]
Total internal reflection, Critical angle
2
Transmitted
(refracted) light
kt
n2
n 1 > n2
ki
1
Incident
light
kr
Reflected
light
(a)
sin  c 
n2
n1
 2  90 
c
Critical angle
(b)
Evanescent wave
1
1   c TIR
(c)
Light wave travelling in a more dense medium strikes a less dense medium. Depending on
 c , which is determined by the ratio of the refractive
the incidence anglewrt to
indices, the wave may be transmitted (refracted) or reflected. (a) 1   c
and total internal reflection (TIR).
(c) 1   c
(b) 1   c
n2
sin  c 
n1
[19]
Phase shift due to TIR
• The totally reflected wave experiences a phase shift however
which is given by:
N
tan

2
n1
n
n2
n 2 cos 2  1  1
n sin  1
;
 p n n 2 cos 2 1  1
tan

2
sin  1
[20]
• Where (p,N) refer to the electric field components parallel or
normal to the plane of incidence respectively.
Optical wave guiding by TIR: Dielectric Slab Waveguide
Propagation mechanism in an ideal step-index optical waveguide.
Launching optical rays to slab waveguide
sin  min 
n2
; minimum angle that supports TIR
n1
[21]
Maximum entrance angle,  0 max is found from
the Snell’s relation written at the fiber end face.
n sin  0 max  n1 sin  c  n1  n2
2
2
[22]
Numerical aperture:
NA  n sin  0 max  n1  n2  n1 2
[23]
n1  n2

n1
[24]
2
2
Optical rays transmission through dielectric slab waveguide
n1  n 2 ;    c 

2
 c
O
For TE-case, when electric waves are normal to the plane of incidence
must be satisfied with following relationship:

2
2
2

 n1 d sin  m   n1 cos   n2
tan 



2  
n1 sin 






[25]
EM analysis of Slab waveguide
• For each particular angle, in which light ray can be faithfully transmitted
along slab waveguide, we can obtain one possible propagating wave solution
from a Maxwell’s equations or mode.
• The modes with electric field perpendicular to the plane of incidence (page)
are called TE (Transverse Electric) and numbered as: TE , TE , TE ,...
0
1
2
Electric field distribution of these modes for 2D slab waveguide can be
expressed as:

Em ( x, y, z, t )  e x f m ( y) cos(ωt   m z )
[26]
m  0,1,2,3 (mode number)
wave transmission along slab waveguides, fibers & other type of optical
waveguides can be fully described by time & z dependency of the mode:
cos(ωt   m z )
or e j (wt   m z )