ECEN 4616/5616
Jan 16, 2013
Light Propagation in materials:
Electromagnetic wave equation for empty space (from Maxwell’s Equations):
 2E
 E   0 0 2  0
t
Comparing to the classical wave equation:
2
1  2Ψ
0
v 2 t 2
we see that EM waves travel at a velocity (in empty space) of
 2Ψ 
𝑣=
1
√(𝜇0 𝜖0 )
≡ 𝑐,
where µ0 is the magnetic permeability and ϵ0 is the electric permittivity. In a
material, these quantities are, in general, different:
𝜇 = 𝜇𝑟 𝜇0 ; 𝜖 = 𝜖𝑟 𝜖0 ,
where the subscript ‘r’ indicates the relative permeability and permittivity. (ϵr is
also known as the ‘dielectric constant’.)
At the atomic scale, magnetic dipoles are (mostly) created by unbalanced current
loops in the outer electron orbitals (unfilled shells), while electric dipoles can be
created by distortion of electron orbitals by an electric field. At optical
frequencies, the atoms cannot move (flip) fast enough to interact with the wave’s
magnetic field, but the electrons do interact significantly with the electric field.
So, for light, 𝜇𝑟 ≈ 1, but 𝜖𝑟 > 1.
Hence the ratio between the speed of light in a medium and the speed in free
𝑐
space is 𝑣 = √𝜖𝑟 ≡ 𝑛, where n is defined as the “index of refraction”.
Also, the wavelength of a wave with frequency f, in a material with index of
𝑣
refraction n is 𝜆 = 𝑓 =
𝑐⁄
𝑛
𝑓
=
𝜆0
𝑛
, where 𝜆0 is the wavelength in free space.
Snell’s Law:
From the relation of the wavelength in a material above, and the requirement
from Maxwell’s equations that tangential fields be continuous across an interface,
it can be shown that the angle of reflection, θr, is the same as the angle of
incidence, θ1, and the relation between the angle of incidence and the refraction
angle, θ2 is 𝑛1 𝑠𝑖𝑛(𝜃1 ) = 𝑛2 𝑠𝑖𝑛(𝜃2 ) , where n1, n2 are the indices of refraction in
the respective media.
Θ1
Θr
n1
(n2 > n1)
n2
Θ2
pg. 1
ECEN 4616/5616
Jan 16, 2013
Total Internal Reflection (TIR):
Suppose, in the above drawing, that n1 > n2 (so that θ2 will be larger than θ1);
Then, there will be some incident angle, θ1 < 90o, for which θ2 = 90o – i.e., there
will be no ray transmitted into material 2. At this point, 𝑠𝑖𝑛(𝜃2 ) = 1, and
𝑛
𝜃1 = 𝑠𝑖𝑛−1 (𝑛2 ). This is known as the critical angle, and any angle of incidence
1
greater will also not result in a transmitted wave.
(For example: if n1=1.5 and n2=1, sin(θ1)=2/3, and θ1=41.8o)
TIR in prisms is used in binoculars (for example) in lieu of mirrors, as 100% of
the light is reflected under TIR conditions – a value that mirrors cannot achieve.
TIR also allows glass rods to transmit light without significant loss, and is a
condition that must be met, even as the rod diameter approaches a few
wavelengths (as in single-mode optical fibers), where wave optics is needed to
fully analyze the transmitted modes.
Form of Surface Equations used in Ray Tracing:
In ray tracing, a common task is to find the intersection of a ray with a spherical
(or otherwise curved) surface. For this purpose, the usual equations of a conic
surface, centered at the coordinate origin is awkward. Instead, we use a form of
the conic sections derived by Karl Schwarzchild in 1905,
y 2  2 Rz  ( K  1) z 2  0 ,
where R is the radius of curvature and K is the shape parameter.
This is a convenient formula because it puts the apex of the curves at the origin
and tangent to the y axis (standard layout for ray tracing), and it combines all the
conic sections into a simple formula.
pg. 2
ECEN 4616/5616
Jan 16, 2013
The shapes for various values of K are:
( y 2  2 Rz  ( K  1) z 2  0 )
Y
2R
Z
Using this formulation, we can solve for the “sag” form of the conic surfaces used
in ray tracing:
1
cr 2
and r is substituted for y (since we are
Z
, where c 
R
1  1  (1  k )c 2 r 2
describing a surface of rotation about the z axis).
The Paraxial Assumption:
Expanding the sag equation about the origin gives the interesting result:
1
1
1
𝑍 = 2 𝑐𝑟 2 + 8 𝑐 3 𝑟 4 (𝑘 + 1) + 16 𝑐 5 𝑟 6 (𝑘 + 1)2 +…,
1
or, just 𝑍 = 2 𝑐𝑟 2 + 𝑂(4)
This tells us that, close enough to the origin, any conic section (indeed, any
smooth surface of revolution) can be considered to be a sphere and whose sag
1
is given by the quadratic, 𝑍 = 2 𝑐𝑟 2, where c ≡ 1/R.
pg. 3
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Jan 16, 2013
Thin Lens Approximation:
A “thin lens” is considered to be a piece of glass (say) with spherical surfaces
(including flat), that cause light to be refracted according to Snell’s Law, but
whose thickness is small enough to neglect. In practice, when using the thin
lens approximation, we consider that rays don’t change their distance from the Zaxis inside the lens – the only effect the lens has is to change the angle of the
ray. (We will drop this approximation when we consider ‘thick’ lenses.)
The effect of a thin lens on an incident plane wave is to delay the wave by (n-1)t,
where n is the lens’ index of refraction (and assuming the surrounding material is
1
1
of index n=1), and 𝑡 = 𝑡0 − 2 𝑐1 𝑟 2 + 2 𝑐2 𝑟 2 (where t0 is the thickness at the Z-axis.)
Note: The above equation uses the “Sign Rules” for optical calculation:
Sign Rules:
1. The (local) origin is at the intersection of the surface and the z-axis.
2. All distances are measured from the origin: Right and Up are positive
3. All angles are acute. They are measured from:
a. From the z-axis to the ray
b. From the surface normal to the ray
c. CCW is positive, CW is negative
4. Indices of refraction are positive for rays going left-to-right; negative for rays
going right-to-left. (Normal is left-to-right.)
On the right side of the lens, the total delay, with respect to the wave’s vertex at
𝑟2
that point, is 𝐷(𝑟) = (𝑛 − 1)(𝑡0 − 𝑡) = (𝑛 − 1) 2 (𝑐1− 𝑐2 ), which we recognize as a
paraxial sphere with radius of curvature, 𝐶 = (𝑛 − 1)(𝑐1 − 𝑐2 ). This wavefront is
1
converging to a point (ignoring diffraction) at a distance 𝑅 = 𝐶 from the lens, so
we have shown that the lens will focus an incident plane wave to a point at this
distance. This distance, F=R, is called the “focal length” of the lens. This gives
us the “Thin Lens Equation”:
1
1
1
= (n − 1)(c1 − c2 ) = (n − 1) ( − )
F
R1 R 2
Where R1, R2 are the radii of curvature of the 1st and second lens surface (using
the sign rules).
pg. 4
ECEN 4616/5616
Jan 16, 2013
Paraxial Ray Tracing:
n
n’
P
i
i’
h
u
u’
a
O
A
z
A’
R
l
l’
Sign Rules:
1. The (local) origin is at the intersection of the surface and the z-axis.
2. All distances are measured from the origin: Right and Up are positive
3. All angles are acute. They are measured from:
a. From the z-axis to the ray
b. From the surface normal to the ray
c. CCW is positive, CW is negative
4. Indices of refraction are positive for rays going left-to-right; negative for
rays going righ-to-left. (Normal is left-to-right.)
Notation Rules:
Unprimed rays, angles and distances are before the ray is refracted by the
surface, primed values are after.
Paraxial Assumptions:
1. Surface sag is ignored as negligible.
2. angles  sines  tangents
Positive Values:


Angles: u, i, i’
Lengths: l’,R,h
Negative Values:


Angles: a, u’
Lengths: l
pg. 5
ECEN 4616/5616
Jan 16, 2013
Angle of incidence & refraction:
i au
 u  a (since u  0, a  0)
i  a  u
 u   a (since u   0, a  0)
Ray Angles with z-axis:
a
u
h
l
u  
h
R


h
R
h
l
h
h

l
l
Snell’s Law (Paraxial):
ni  ni
nu  a   nu  a 
 h h
 h h
n     n   
 l R
 l R 
n
n
1
  nc    nc , where c 
l
l
R
n n
  cn  n   K  Imaging equation for a surface
hence: l  l
Where, K  c(n  n) is defined as the surface power
Thin Lens Equation:
The thin lens equation can be derived by applying the above surface imaging
equation sequentially to both surfaces of the thin lens.
Note that a thin lens is defined as two surfaces with negligible separation
(both surfaces intersect z-axis at origin), considered paraxially.
We will use subscripts 1, 2 to indicate which surface we are tracing through.
Note that the resultant values (primed) from surface one are the input
(unprimed) values for surface two: i.e., l1  l 2 , etc.
pg. 6
ECEN 4616/5616
Jan 16, 2013
Using the surface imaging equation twice, we have:
n1 n1
  c1 n1  n1   K 1
l1 l1
n2 n2

 c 2 n2  n2   K 2
l 2 l 2
l1  l2 ,
and setting :
Noting that n1  n2  1 (air)
n1  n2  n (the glass index)
l1  l , l2  l , (the input and output of the thin lens)
We get, after substitution:
n 1
  c1 n  1  K1
l2 l
1 n
  c2 1  n   K 2
l  l2
Adding
these two equations gives:
1 1
  n  1c1  c2   K1  K 2
(Paraxial thin lens equation)
l l
(Notice that this also gives us the relation between the focal length, object
and image:
1 1 1
= − ,
𝐹 0 𝐼
where F is the focal length, O is the distance to the object and I the distance
to the image.
K
Observations about the thin lens equation:
1. l, l’ are independent of h – all rays from A go to A’ regardless of
(paraxial) height on the lens: => A’ is the (paraxial) image of A
2. We can use this equation without regard to the sign of the values, or
whether the image is real or virtual, as long as the sign conventions
are followed.
3. The power of the lens is the sum of powers of each surface. This will
be true for any number of (paraxially) close surfaces – which will be
used in deriving achromats.
pg. 7
ECEN 4616/5616
Jan 16, 2013
Paraxial Refraction and Transfer ray tracing:
The thin lens equation allows us to trace images through a series of surfaces or
thin lenses, but doesn’t allow the path of a specific ray to be traced. Hence, we
need a set of equations for this task that don’t involve the object and image
distances, l,l’.
From the Paraxial Snell’s law, and making some of the same substitutions as
above:
ni  ni
nu  a   nu   a 
h
h


n u    n u   
R
R


nu  hc   nu   hc 
nu   nu   hcn  n    hK
Hence nu  nu  hK is the Gaussian Refraction Equation for a surface whose
1
power is defined as cn  n  K . Remember that c 
, where R is the center
R
of curvature of the surface, whose magnitude and sign are given by the sign and
coordinate conventions, and n, u are the index and angle before (to the left of)
the surface, n’, u’ are the quantities after (to the right of) the surface.
The Gaussian Refraction Equation gives you the change in angle as a ray with
height h passes through a surface, allowing arbitrary rays to be drawn.
There is one more step necessary to trace a ray through multiple surfaces, and
that is a transfer equation that allows you to calculate the change in height of a
ray as it travels to the next surface.
Important note on notation:
The distance from a surface to the next surface is considered a property of the
first surface: Hence the distance from surface S1 to S2 is labeled d1, not d2. This
is a convention not only in paraxial tracing, but in all ray trace programs, as well.
There are two transfer equations, whose derivation is self-evident (provided the
sign conventions are carefully followed):
For the imaging equation, the output image distance for surface i, adjusted by the
distance between surfaces, becomes the input object distance for surface i+1.
li 1  li  di
For the refraction equation, the output height modified by the output slope times
d gives the input height for the next surface:
hi 1  hi  diui
pg. 8
ECEN 4616/5616
Jan 16, 2013
Use of paraxial equations for finite height rays:
Notice that the paraxial formulas (refraction, transfer) for Gaussian tracing are all
exact formulas, regardless of finite ray height, IF:
1. “Angle” variables, u, u’ are interpreted to be the tangents of the actual ray
angles.
2. “Lenses” and “surfaces” are considered to be planes perpendicular to the
z-axis at the local coordinate origin.
3. “Paraxial Lenses” have no curvature or sag, only power.
4. All refraction takes place at the intersection of a ray with the surface
plane.
5. The variables u, u’ are modified by the surface according to the Gaussian
refraction equation – Snell’s law is ignored.
Under these conditions, the Gaussian trace through a “paraxial” surface
simulates the performance of an ideal lens – one which images all rays in exactly
the same place as paraxial rays, regardless of the distance from the axis that a
ray intersects the surface.
All ray trace programs have a “Paraxial Lens” type which behaves as such an
ideal lens. One simply specifies a focal length (1/K). There is no need to restrict
rays to the actual paraxial region.
pg. 9
ECEN 4616/5616
Jan 16, 2013
ECEN 4616/5616
Using Gaussian ray traces to generate lens formulas:
Trace a horizontal, marginal ray through two lenses (why a horizontal ray?):
n1
n1’=n2
n2’
u1=0
h1
u1’=u2
u2’
h2
f
d1
K1
K2
Note that: u2’ = -h1/f
(definition of f for the
combination)
Refract through first element:
n1u1  n1u1  h1 K1
u1  
h1 K1
, since u1  0
n1
h1 K1
, u 2  u1 , n2  n1
n2
Transfer to second element:
h2  h1  u1d1
 u2  
h1 K1
d1
n2
Refract at second element:
n2 u 2  n2 u 2  h2 K 2
 h2  h1 
 hK  

hK
 n2   1 1    h1  1 1 d1  K 2
n2
 n2  

 h 
KK
n2   1    h1 K1  h1 K 2  h1 1 2 d1
n2
 f 
pg. 10
ECEN 4616/5616
Jan 16, 2013
1
where K is the combination power, we get the:
f
General formula for combination of two systems.
Eliminating h1 and using K 
n2 K  K1  K 2 
K1 K 2
d1
n2
(Easy to remember & independent of the sign
convention)
Why doesn’t n1 enter into this equation?
(n1 is included in the calculation of K1)
Thick Lens Equation:
n2  1, n  n2 , d  d1
Let: K1  c1 n  1 (power of 1st surface)
K 2  c2 1  n  (power of 2nd surface)
K  c1 n  1  c2 1  n   c1c2
n  11  n d
Then:
n  1 d 

 n  1c1  c2  c1c2

n

Or, in terms of focal length and radii:
1
1
1 n  1 
 n  1 

d 
f
R
R
nR
R
2
1 2
 1

n
Thick lens formula (Lensmaker’s formula)
Thin Lens Equation:
Just set d=0 in the thick lens equation:
1
1
1 
 n  1  
f
 R1 R2 
Focal length of lenses not in air:
Suppose that n2’ 1 (lens is not in air):
Then, the LHS of the lens equations becomes
n 2
, and K2 will be calculated
f
differently.
pg. 11
ECEN 4616/5616
Jan 16, 2013
Two thin lenses spaced in air:
n2  n2  1
Let:
K1 
Then:
1
1
, K2 
f1
f2
1
1
1
d
 

f
f 1 f 2 f1 f 2
(This also works for thick lenses, if d is measured between principle planes)
Afocal systems:
Suppose that K1K 2 d  K1  K 2
K  0,  f  
Then
K  K2
d 1

K1 K 2
1
f1
1
 1
f2
 f1  f 2
f1 f 2
Hence if the lenses are spaced by the sum of their focal lengths, we get an afocal
system – e.g., a telescope.
Zoom lens:
What happens if d = f1 ?
Then f = f1 and the second lens has no effect on the total power at all.
This is the principle behind zoom lenses: The power contribution of a lens (or
lens group) to the total lens is dependent on the relative positions of the lenses.
pg. 12