Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Lecture 1: Mathematical Basis of Optical Fields and Eikonal Equations
(06/24/13)
Outline:
A. Introduction
B. Summary of Maxwell’s Equations
o Light in a medium: need for Substitutive relationship
C. Maxwell’s Equations in Cartesian Coordinates
o Propagation of source-free EM wave in homogeneous medium
o Field associated with a line current source
D. Geometrical light rays
E. Path of Light in an Inhomogeneous Medium
F. Fermat’s Principle of least time
A. Introduction:
While Maxwell’s equations can solve light propagation in a rigorous way(say using your
COMSOL or FDTD/FEM software), the exact solutions can be found in fairly limited cases,
and most practical examples require intuitive approximations (so we can take back-of
envelope estimation!).
Distance of event
wavelength
Geometric
Optics
Radio
Engineering,
(Scalar) Fourier Optics
Nano-Optics
Vector Field, Polarization
Antennas,
Transmission
lines,
cavities,
amplifiers
Wavelength
min feature size
Figure 1. Different domains and approaches of EM waves
For example, radios and mobile phones make use of same Maxwell Equations to
transfer information as carried by light waves, but our perception is quite different. Why?
We tend to think of light as bundles of rays in our daily life. This is because we observe
the processes (emission, reflection, scattering) at a distance (> 10cms with bare eyes) that
are much longer than the wavelength of light (10-7m or 400-700nm), and our receivers
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
(retina and CCD pixels) are also considerably large. In the other end, the wavelength of
radio-frequency waves (10cm at 3GHz) is comparable or sometimes larger than the size
and spacing between transmitting/receiving devices (say, the antennas in your cell
phones).
Based on the specific method of approximation, optics has been broadly divided into
two categories, namely:
i.
Geometrical Optics (ray optics) treated in the first half of the class;
emphasis on finding the light path; it is especially useful for:
-
Designing optical instruments;
or tracing the path of propagation in inhomogeneous media.
ii.
-
Wave Optics (physical optics).
Emphasis on analyzing interference and diffraction
Gives more accurate determination of light distributions
B. Summary of Maxwell Equations, Differential Forms:
Symbols
E
H
D
B

Physical Quantity
Electric Field
Magnetic Field
Electric flux density
Magnetic flux density
Volume Charge Density
Current density
Permittivity of Free Space
Permeability of Free Space
J
0
0
Units (in real space)
Volts/m
Amps/m
Coulombs/m2
Tesla
Coulombs/m3
Amps/m2
8.85x10-12 Farads/m
4x10-7 Henry/m
1. In real space, time-dependent fields:
(Faraday’s Law:)
(B1)
(Ampere’s Law:)
(B2)
(Gauss’ Law, electric field)
(Gauss’ Law, magnetic field)
Note: J, q are sources of EM radiation and E, D, H, B are induced fields.
(B3)
(B4)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
2. From time domain to frequency domain:
Continuous wave laser light field understudy are often mono-chromatic. These
problems are mapped in the Maxwell equations by expanding complex time
signals to a series of time harmonic components (often referred to as “single”
wavelength light):
⃗( ) ∫ ⃗(
)
(
)
e.g.
(B5)
Advantage:
⃗ (⃗
)
∫ ⃗⃗ (⃗
)
(
)
⃗ (⃗
∫ [
)]
(
)
(B6)
So, we can replace all time derivatives
by
in frequency domain:
(Faraday’s Law:)
(B7)
(Ampere’s Law:)
(B8)
The forms of the two Gauss’ Law remain unaltered.
3. From real space to wave-vector space:
Similarly, we can simplify the problems by expanding complex spatially varying
signals to a series of spatial harmonic components:
⃗ (⃗
e.g.
Advantage:
⃗⃗ (⃗
(
)
)
∫
)
(
(
)
∫
⃗[⃗
)
⃗ ⃗ (⃗
∫
⃗ ⃗ (⃗
⃗⃗ (
)]
)
)
( ⃗ ⃗)
(B9)
( ⃗ ⃗)
(B10)
( ⃗ ⃗)
Lecture Notes on Short Course on Nanophotonics,
So, we can replace all space derivatives
by
prepared by Nick Fang
⃗ in wave-vector domain:
(Faraday’s Law:)
(B11)
(Ampere’s Law:)
(B12)
(Gauss’ Law, electric field)
(B13)
(Gauss’ Law, magnetic field)
(B14)
These set of equations are particularly helpful when thinking about propagation
and focusing of white light or signal of broad frequencies, while the spatial
variation is significant (e.g. gratings and nanoparticles).
4. Complete transform: frequency and wavevector space representation
Finally, the problem understudy can be written with a joint transformation of
space and time (say, a holographic grating under the illumination of a HeNe
Laser):
⃗ ⃗ (⃗ )
⃗ (⃗ )
e.g.
(⃗ ⃗
)
(B15)
∫
(
By replacing
by
⃗ and
)
by
we arrive at:
(Faraday’s Law:)
(B16)
(Ampere’s Law:)
(B17)
(Gauss’ Law, electric field)
(B18)
(Gauss’ Law, magnetic field)
(B19)
Note: As you can see in this example, B field is orthogonal to E and k (often
considered as direction of propagation), while D is orthogonal to H and k when
there is no source.
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
- Observation: Light in a medium -Need for Substitutive relationships
There are total of 12 unknowns (E, H, D, B) but so far we only obtained 8 equations
from the Maxwell equations (2 vector form x3 + 2 scalar forms) so more information
needed to understand the complete wave behavior!
Generally we may start to construct the response of a material by applying a
excitation field E or H in vacuum. Therefore it is more typical to consider the E, H
field as input and D, B fields as output. Some generic form of such equations could
be written as:
(
)
(B20)
(
)
(B21)
Note that such response could be both dependent on space and time, ie. Cumulating
contributions from the reacting field in the neighborhood and could experience
delay to transfer energy from one form to the other. They are generally not linear,
and in addition, the response is not isotropic (D E, and B H). Those properties will
be revisited further in nonlinear optics, plasmonics and metamaterials.
In common optical materials, we may enjoy the following simplification of local (i.e.
independent of neighbors) and linear relationship:
⃗⃗ ( )
( ) ⃗( )
(B20)
⃗⃗ ( )
( ) ⃗⃗⃗ ( )
(B21)
The so called (electric) permittivity ( ) and (magnetic) permeability ( )are
unitless parameters that depend on the frequency of the input field. In the case of
anisotropic medium, both ( ) and ( )become 3x3 dimension tensor.
Now we have 6 more equations from material response, we can include them
together with Maxwell equations to obtain a complete solution of optical fields with
proper boundary condition.
C. Maxwell’s Equations in Cartesian Coordinates:
To solve Maxwell equations in Cartesian coordinates, we need to practice on the
vector operators accordingly. Most unfamiliar one is probably the curl of a vector .
In Cartesian coordinates it is often written as a matrix determinant:
Lecture Notes on Short Course on Nanophotonics,
̂
⃗
̂
prepared by Nick Fang
̂
|
|
(C1)
In this fashion, we may write the Faraday’s law in frequency domain,
, with 3 components in Cartesian coordinates:
(
)
(C2)
(
)
(C3)
(
)
(C4)
Likewise, we arrive at the rest of Maxwell’s equations:
(
)
(C5)
(
)
(C6)
(
)
(C7)
together with
(C8)
And
(C9)
You may compare the above with equations (2.6a-f) of Maier’s textbook chapter 2.
- Example 1: Propagation of source-free EM wave in 1-D homogeneous medium
(e.g. an expanded laser beam in +x direction,
)
We can now further simplify from the above equations in Cartesian coordinates:
(
)
(C10)
(
)
(C11)
(
)
(C12)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
And
(
)
(C13)
(
)
(C14)
(
)
(C15)
From the above we found only 4 non-trivial equations. In the isotropic case, they can
be further divided into 2 independent sub-groups (two Polarizations!):
(Ez, Hy only)
(C16)
and
(C17)
Or (Ey, Hz only)
(C18)
And
(C19)
Observations:
- Once again we see that wave propagation in such medium is purely transverse,
i.e. only components of E, H field that are orthogonal to propagation direction
(+x) survived in the wave field.
- If we know the prescribed source (e.g. an oscillating current Jy or Jz at x=0
position) then we can plot the complete spatial distribution of optical field at
arbitrary x location.
-
Taking the derivative
again on any of these equations, we obtain wave
equation such as:
(
(
)
(
)
(C20)
)
(C21)
since the speed of light c0 in vacuum satisfy (
Therefore the index of refraction is found as: ( )
-
)
√ ( ) ( )
In the k-presentation, there is a linear relation between E and H component,
similar to ohm’s law, e.g.
.
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
We take the ratio of Ez/Hy (since they have the unit of voltage divided by current)
)in the medium:
to define wave impedance (
(
)
(C22)
You may verify that in vacuum (
)
Z 0=
=377 


(C23)


In general however, wave impedance is a function of polarization, material
property, and longitudinal wavevector kx.
-
Example 2: Field associated with a line current source, and it’s connection
to Evanescent waves
y
̂
( ) ( )
x
∫
(
)
=∫
Figure 2: Measurement of 2D H Field at arbitrary plan y>0, excited by a line current
source J= ̂ ( ) ( ) placed at the origin.
In previous examples we studied the field associated with a flat interface in which the
evanescent waves can be excited by a plane wave illuminating from the dense medium. But
how is it related to imaging? How is the analysis from beams connected to imaging an
object of arbitrary shapes?
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
)
In this analysis, we take the example of a line of current sources ( (
in 2 dimension) and study the field excited by the line source in vacuum.
̂
( ) ( )
Let’s begin by the Maxwell equations of k- domain and in vacuum:
⃗
⃗
⃗
⃗
⃗
(C24)
⃗
(C25)
⃗ (
⃗)
(C26)
⃗ (
⃗)
(C27)
In order to find E(k,) as a function of source J(k,)= ̂ , we can apply ⃗
⃗
(⃗
⃗)
(
⃗
⃗)
⃗ (⃗
on (C24):
⃗ (⃗ ⃗ )
⃗)
⃗
(C28)
(C29)
Therefore, when we assemble the terms of E(k,) together,
⃗
⃗ (⃗
⃗
⃗)
(C30)
The right hand is a source term, while the left hand side is still complex as we have a term
of ⃗ ⃗ projected to k direction.
Using equation (C26) we see that ⃗ ( ⃗
⃗ ( ) so the term indicates a fluctuating
⃗)
charge associated with the current source. Now we apply conservation of charges
(
)in k- domain:
⃗
⃗ (⃗
)⃗
(
Or ⃗
Likewise, if you started by ⃗
(C31)
(
)
)
⃗ (⃗ )
̿
(C32)
(C33)
(C25), you will find H(k,) from J
Or ⃗
⃗
)⃗
(
(
)
⃗
(C34)
(C35)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
In Cartisian coordinates, we arrive at:
(
(
)
(
)
)
(
(C36)
)
(C37)
Until now, we allowed the wavevectors k=(kx, ky) to take arbitrary values and independent
of each other. But how does that translate into propagating and evanescent waves? Now
let’s select one direction, say y direction, and perform inverse Fourier transform of H field
on this direction:
(
)
(
(
)
(
(
)∫
)
(
(C38)
)
(
)∫
)
(
(C39)
)
Now we need to evaluate the integral that contains a fast oscillating field
nominator, and a function in the denominator, (
roots:
(
) in the
) which contains 2
)
√
(
(
)(
)
can be a complex number.
Take Eq(C39) as an example, we now can split the integral into two parts:
(
(
)
)
(
{∫
)
(
(
∫
)
)
(
)
}
(C40)
For each of the term to be integrated we can now apply Cauchy’s integral theorem:
( )
∫
(
)
( )
(
(
)
{
)
(
)
(
)}
(C41)
Case I:
When
, then the above forms describe 2 propagating waves:
(
(
)
√
)
{
(
√
)
( √
)}
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Case II:
, we see A=i√
When
=iand to keep the field from
divergence at infinity we eliminate the exponentially growing term, so the above equation
(C41) defines a field that is exponentially decaying away from the source:
(
)
(
)
√
{
( | |√
)}
(C42)
Observations:
Propagate to
new y plane
y
x
(
(
)
)
Figure 3: k- domain analysis of H field excited by the line current source J= ̂
-
The field excited by the objects (such as a single fluorescent molecule that radiate at
the origin x=y=0) can be now considered as a set of beams, separated by their
corresponding lateral wavevector kx. At arbitrary distance y from the object we see
their amplitudes are determined by
-
( ) ( ) .
;
If the period of oscillation at the source is small, kx is large, the amplitude we can
detect is then diminishing exponentially at distance comparable to
√
.
(C43)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
-
Connection to Fresnel Equations: when the source is placed next to an interface,
then we can think of the radiation being modified by inclusion of a set of reflected
and transmitted beams across the interface. The reflection and transmission
coefficients of each kx components are determined by the Fresnel equations, or
based on the mismatch of impedance across the interface.
-
Imaging an object involves collection of the set of reflected and/or transmitted
beams at a distance from the object. Unfortunately we see that a portion of
information associated with
is lost when we are far away from
the object. In order to capture that portion of information we have to move the
interface close enough to the object (i.e. to measure separation better than
then we have to sit at a distance
,
) Also, as the evanescent waves
√
travels along the interface (at x direction), we may need to find methods to capture
these evanescent waves sideways (such as creating curvatures).
D. Propagation of Phase Front and High Frequency Limit, connection to
Geometric Rays
So far we have examined one aspect of nanophotonics, that is, how to analyze the
field near a source, typically in a dimension comparable or smaller than the
corresponding wavelength. In practice however, not all dimensions are equally
small(say, the field is varying rapidly in z, but changes rather slowly in x,y
directions). Intuitively we tend to the approach of Geometric optics such as ray
tracing.
How can we obtain such picture from Maxwell’s equations? Now let’s go back to
real space and time-frequency domain (in a source free, isotropic medium but with
spatially varying permittivity ( ), for example).
⃗
⃗
⃗
(
(
(D1)
( )⃗
( )⃗ )
⃗)
(D2)
(D3)
(D4)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Now we decompose the field E(r, ) into two forms: a fast oscillating component
exp(ik0),
and a slowly varying envelope E0(r) as illustrated in the
following diagram:
Slowly varying envelope
E0(r)

Figure 4: Example of decomposition of E field into the product of slowly varying envelope
and a fast oscillating phase exp(
)
Likewise, we can treat H(r, ) in similar fashion:
⃗⃗⃗
⃗⃗⃗⃗⃗ ( )
⃗⃗⃗⃗⃗ ( )
(
(
)
And
⃗
)
(
)
⃗⃗⃗⃗⃗ ( )
⃗
⃗⃗⃗⃗⃗ ( )
(
)
(
)(
)
⃗⃗⃗⃗⃗ ( )
⃗
⃗⃗⃗⃗⃗ ( )
(
)
(
)(
) ⃗⃗⃗⃗⃗ ( )
With this treatment, we can rearrange the Maxwell equations into:
⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗ )
( )(
(
)
( ) ⃗⃗⃗⃗
⃗⃗⃗⃗⃗
⃗⃗⃗⃗
(D5)
( )⃗⃗⃗⃗
(
)
(D6)
( )⃗⃗⃗⃗
(D7)
(D8)
Furthermore, if the envelope of field varies slowly with wavelength (as we can see in
systems with small loss:
Lecture Notes on Short Course on Nanophotonics,
⃗⃗
⃗⃗⃗⃗ ( )(
(
prepared by Nick Fang
)
),
then to the lowest order in 1/k0 we obtain:
⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗
(D9)
( )⃗⃗⃗⃗
(D10)
(
) ⃗⃗⃗⃗
(D11)
(
) ⃗⃗⃗⃗
(D12)
Equations (D11) and (D12) simply suggests that E0, H0 are orthogonal to the gradient of
phasefront ( ).
E0
H0
2
1
3
Figure 5. Geometrical Relationship of E, H, and
Also, taking
(D9) we obtain:
(
⃗⃗⃗⃗ )
(
⃗⃗⃗⃗ )
( )⃗⃗⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗ |
⃗⃗⃗⃗ |
|
|
( )⃗⃗⃗⃗
Thus for non-zero envelop field E0 we have
|
|
( )
(D13)
Lecture Notes on Short Course on Nanophotonics,
|
|
( )
prepared by Nick Fang
( )
(D14)
In Cartesian Coordinates, we can write (D14) as:
(
)
(
)
(
(
)
This is the well-known Eikonal equation,
word, meaning image).i
-
)
(D15)
being the eikonal (derived from a Greek
Observation (not proof):
The above equation yields: |
|
, or |
|
This is equivalent to the Fermat’s Principle on optical path length (OPL):
∫|
|
∫
(D16)
Such process requires the direction of the light path ⃗⃗⃗ follows exactly the gradient
of phase contour
(a vector). We will use it to determine the path of light in a
general inhomogeneous medium.
E. Path of Light in an Inhomogeneous Medium
-
Example 1: 1D problems (Gradient index waveguides, Mirage Effects)
Figure 6. The mirage effect
The best known example of this kind is probably the Mirage effect in dessert or
near a seashore, and we heard of the explanation such as the refractive index
increases with density (and hence decreases with temperature at a given altitude).
With the picture in mind, now can we predict more accurately the ray path and
image forming processes?
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Starting from the Eikonal equation and we assume
then we find:
(
)
(
)
(
) is only a function of x,
( )
(D17)
Since there is the index in independent of z, we may assume the slope of phase
change in z direction is linear:
(
)= C(const)
√
( )
(D18)
This allows us to find
(D19)
From Fermat’s principle, we can visualize that direction of rays follow the gradient of
phase front:
(D20)
z-direction:
x-direction:
( )
(D21)
√
( )
( )
(D22)
Therefore, the light path (x, z) is determined by:
√
(D23)
( )
Hence
∫
-
√
( )
(D24)
Example: In a fish tank with sugar solution at the bottom, we observed that due to
the index variation produced by a concentration gradient of dissolved sugar, some
light rays follow a curved path instead of straight lines. Based on mass diffusion
equations, we observed that the index of refraction in the sugar solution is
approximately:
Lecture Notes on Short Course on Nanophotonics,
( )
[
(
prepared by Nick Fang
)].
(D25)
Where, n0 is the index of refraction of water (1.34), is the index change in
saturated sugar solution (~0.14), D is the diffusivity of sugar in aqueous
solution(D~6x10-10 m2/s), and t is the time the solution is prepared (~3 days).
If we only consider rays very near the bottom of the tank, we can expand the
exponential term in the denominator around x = 0:
e

x2
2 Dt
1
x2
x4


2 Dt 4Dt 2
(D26)
Without loss of generality, we may assume a quadratic index profile along the x
direction, such as found in gradient index optical fibers or rods:
( )
∫
(
)
(D27)
(D28)
√ (
)
To find the integral explicitly we may take the following transformation of the
variable x:
√
(D29)
Therefore,
∫
(D30)
√
√
(
)
(D31)
Or more commonly,
√
(
√
(
))
(D32)
As you can see in this example, ray propagation in the gradient index waveguide follows a
sinusoid pattern! The periodicity is determined by a constant
√
.
Lecture Notes on Short Course on Nanophotonics,
Index of
refraction
n(x)
prepared by Nick Fang
x
dz
dx
z
Figure 7. The ray path in a gradient index slab
Observation: the constant C is related to the original “launching” angle  of the
optical ray. To check that we start by:
|
If we assume C= ( )
√
(
(D33)
)
, then
|
(D34)
- Case II: axisymmetric, cylindrical gradient index materials
In some cases such as the famous Maxwell Fisheye lens and Luneberg lens, the index of
( ) is only a
refraction is varying in a centro-symmetric fashion. Thus we assume
function of r in Equation (D14):
(
)
(
)
( )
(D35)
A there is no dependence of  in the index, we may assume the slope of phase
change in  direction is linear too:
(
)
(D36)
√
( )
(D37)
Once again we can visualize that direction of rays follow the gradient of phase front
( ), so (B36) and (B37) can be expressed as:
( )
(D38)
(This is tricky!)
( )
√
( )
(D39)
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Therefore, the light path (r, ) is determined by:
(D40)
√
( )
∫
(D41)
( )
√
Note: The general result of centro-symmetric gradient index lens is consistent with
Literature (M. Born & E. Wolf, Principles of Optics, 7th ed., p. 130, 157-158)
Example: Maxwell’s fish-eye lens
A hypothetical “fish-eye” lens is investigated mathematically by Maxwell as follows.
Note this lens is under hot debate recently whether it promises a “perfect” focus.ii
( )
(
(
∫
(D42)
)
)
√
(
(D43)
)
,
Where
The explicit solution of (D43) is provided by Born and Wolf:
{√
}
{√
}
(D44)
Or,
(
where
(r0, ).
)
√(
{√
)
(D45)
} indicates a “launching” angle at the initial point
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
Observation:
-
To obtain the rays of such a lens of Equation(D45) into Cartesian coordinates, we
use the transformation:
,
:
(
)
)
√(
(
(
√(
√(
)
)
)
(D46)
)
)
√(
(
√(
(
)
)
)
(
)
.
Equation (D47) clearly indicates each ray forms a circle with radius
-
(D47)
The RHS of Equation (D45) is only a function of r. An interesting result is, if we
replace any value r by
, then we only need to multiply the RHS by (-1). This is
achievable at the LHS side if we simply ask
. Therefore, all rays leaving a
point P(r0, 0) will refocus back to a point Q(
with respect to P, and the magnification is
), forming an inverted image
.
P (r0, 0).
a
r=a
Q

Figure 8. Ray Schematics of Maxwell’s Fish Eye Lens with a radially varying index of
refraction described by (D35). All rays (blue solid curves) from point P will follow
circular path indicated by Eq(D47), and focus to a point Q (
).
Lecture Notes on Short Course on Nanophotonics,
-
prepared by Nick Fang
Other popular examples: Luneberg Lens
The Luneberg lens is inhomogeneous sphere that brings a collimated beam of light to a
focal point at the rear surface of the sphere. For a sphere of radius R with the origin at
the center, the gradient index function can be written as:
( )
{
√
(D48)
Such lens was mathemateically conceived during the 2nd world war by R. K. Luneberg,
(see: R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence,
Rhode Island, 1944), pp. 189-213.) The applications of such Luneberg lensiii was quickly
demonstrated in microwave frequencies, and later for optical communications as well
as in acoustics. Recently, such device gained new interests in in phased array
communications, in illumination systems, as well as concentrators in solar energy
harvesting and in imaging objectives.
Figure 9, Left: Picture of an Optical Luneberg Lens (a glass ball 60 mm in diameter)
used as spherical retro-reflector on Meteor-3M spacecraft. (Nasa.gov)
Right: Ray Schematics of Luneberg Lens with a radially varying index of refraction. All
parallel rays (red solid curves) coming from the left-hand side of the Luneberg lens will
focus to a point on the edge of the sphere.
F. Fermat’s Principle of least time
At first glance, Fermat’s principle is similar to the problem of classical mechanics:
finding a possible trajectory of a moving body under a given potential field (We will
introduce such Lagrangian or Hamiltonian approach in more detail in the coming lectures
about gradient index optics).
Lecture Notes on Short Course on Nanophotonics,
prepared by Nick Fang
The underlying argument is, light propagating between two given points P and P’,
would take the shortest path (in time). In order to quantify the variation of light speed in
different medium, we introduce of an index of refraction n:
Where: c~3x108 m/s is speed of light in vacuum;
and v is the speed of light in the medium.
Using the index of refraction, we can define an “Optical Path Length”(OPL):
( )
∫
( ⃑)
(F1)
This is equivalent to finding the total time (T= OPL/c) required for signals to travel from P
to P’, and vice versa.
How is it consistent with wave picture? Modern theorists like Feynman take more
rigorous approach to show that all other paths that do not require an extreme time
((shortest, longest or stationary) are cancelled out, leaving only the paths defined by
Fermat’s principle.
Endnotes and References
For reference about the high frequency (also coined as Physical Optics) limit of
Maxwell equations, you may read:
i
-Giorgio Franceschetti, Chapter 5: “High Frequency Fields” in “Electromagnetics:
Theory, Techniques, and Engineering Paradigms”, Plenum press, 1997.
ii
See references as following:
-
U. Leonhardt, “Perfect imaging without negative refraction”, New J. Phys. 11, 093040
(2009).
-
Y.G. Ma, S. Sahebdivan, C.K. Ong, T. Tyc, and U. Leonhardt, “Evidence for
subwavelength imaging with positive refraction”, New. J. Phys. 13, 033016 (2011).
-
Juan C González et al , “Perfect drain for the Maxwell fish eye lens”, New J. Phys. 13
023038 (2011).
Lecture Notes on Short Course on Nanophotonics,
iii
prepared by Nick Fang
-
R J Blaikie, “Perfect imaging without refraction?” New J. Phys. 13 125006 (2011).
-
Xiang Zhang, "No drain, no gain"(Comment), Nature, 2011))
References regarding Luneberg Lens:
-
S. P. Morgan,“General Solution of the Luneberg Lens Problem”, J. Appl. Phys. 29,
1358 (1958); doi: 10.1063/1.1723441
-
F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381
(1974).
-
S. K. Yao and D. B. Anderson, “Shadow sputtered diffraction-limited waveguide
Luneburg lenses,” Appl. Phys. Lett. 33, 307–309 (1978).
-
S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, and C. M. Oania, “Guided-wave
optical thin-film Luneburg lenses: fabrication technique and properties,” Appl. Opt.
18, 4067–4079 (1979).
-
E. Colombini, “Design of thin-film Luneburg lenses for maximum focal length
control,” Appl. Opt. 20, 3589–3593 (1981).
-
E. Colombini, “Index-profile computation for the generalized Luneburg lens,” J. Opt.
Soc. Am. 71, 1403–1405(1981).