Short Course Lecture 1 EM Waves and Eikonal Equations

7/24/2019 Short Course Lecture 1 EM Waves and Eikonal Equations

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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 Maxwells Equations

o Light in a medium: need for Substitutive relationship

C. Maxwells 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. Fermats Principle of least time

A.

Introduction:

While Maxwells 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!).

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

Vector Field, Polarization

(Scalar) Fourier Optics

Geometric

Optics RadioEngineering,

Antennas,

Transmission

lines,

cavities,

amplifiers

Nano-Optics

Wavelength

min feature size

Distance of event

wavelength


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(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 intotwo 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 Physical Quantity Units (in real space)

E Electric Field Volts/m

H Magnetic Field Amps/m

D Electric flux density Coulombs/m2

B Magnetic flux density Tesla

Volume Charge Density Coulombs/m3

J Current density Amps/m2

0 Permittivity of Free Space 8.85x10-12Farads/m

0 Permeability of Free Space 4x10-7Henry/m

1.

In real space, time-dependent fields:

(Faradays Law:) (B1)

(Amperes Law:) (B2)

(Gauss Law, electric field) (B3)

(Gauss Law, magnetic field) (B4)

Note: J, q are sources of EM radiation and E, D, H, Bare induced fields.


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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 derivativesby in frequency domain:


(Amperes 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. ( )( ) (B9)Advantage: ( ) (B10) [ ]( )


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So, we can replace all space derivatives by in wave-vectordomain:(Faradays Law:) (B11)

(Amperes Law:) (B12)



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:


(Amperes Law:) (B17)



Note: As you can see in this example, Bfield is orthogonal to Eand k(often

considered as direction of propagation), while Dis orthogonal to Hand kwhenthere is no source.


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- 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 (DE, andB H). Those properties willbe 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 areunitless parameters that depend on the frequency of the input field. In the case ofanisotropic 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.

Maxwells 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:


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(C1)In this fashion, we may write the Faradays law in frequency domain,

, with 3 components in Cartesian coordinates: (C2) (C3) (C4)Likewise, we arrive at the rest of Maxwells equations:

(C5) (C6) (C7)together with (C8)And

(C9)

You may compare the above with equations (2.6a-f) of Maierstextbook 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)


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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, Hyonly) (C16)and (C17)

Or (Ey, Hzonly) (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 Jyor Jz at x=0

position) then we can plot the complete spatial distribution of optical field atarbitrary x location.

-

Taking the derivative again on any of these equations, we obtain wave

equation such as: (C20) (C21)since the speed of light c0in 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 ohms law, e.g. .


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We take the ratio of Ez/Hy(since they have the unit of voltage divided by current)

to define wave impedance in the medium:

(C22)

You may verify that in vacuum ( )Z0==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 its connection

to Evanescent waves

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?

y

x ()()

()

=


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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.

Lets 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 fluctuatingcharge associated with the current source. Now we apply conservation of charges( )in k-domain:

(C31) (C32)

Or

(C33)

Likewise, if you started by (C25), you will find H(k,) from J (C34)Or (C35)


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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

lets select one direction, say y direction, and perform inverse Fourier transform of Hfield

on this direction:

() (C38)

()

(C39)

Now we need to evaluate the integral that contains a fast oscillating field in thenominator, and a function in the denominator, ( )which contains 2roots:

( ) 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 Cauchys integral theorem:

(C41)

Case I:

When , then the above forms describe 2 propagating waves:


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Case II:

When , we see A=i=i and to keep the field fromdivergence 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:

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, kxis large, the amplitude we can

detect is then diminishing exponentially at distance comparable to

. (C43)

y

x

() ( )

Propagate to

new y plane


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-

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 kxcomponents 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 fromthe 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 wavestravels along the interface (at x direction), we may need to find methods to capturethese 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 Maxwells equations? Now lets 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)


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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 thefollowing diagram:

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:

E0(r)

Slowly varying envelope


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,then to the lowest order in 1/k0we obtain:

(D9)

(D10) (D11) (D12)Equations (D11) and (D12) simply suggests that E0, H0are orthogonal to the gradient of

phasefront .

Figure 5. Geometrical Relationship of E, H, and Also, taking (D9) we obtain:

( ) ( ) ||

||

Thus for non-zero envelop field E0 we have|| (D13)

1

2

3

E0H0


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|| (D14)In Cartesian Coordinates, we can write (D14) as:

(D15)

This is the well-known Eikonal equation, being the eikonal (derived from a Greekword, meaning image).i

-

Observation (not proof):

The above equation yields: || , or || This is equivalent to the Fermats Principle on optical path length (OPL):

|| (D16)

Such process requires the direction of the light path follows exactly the gradientof phase contour (a vector). We will use it to determine the path of light in ageneral 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?


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Starting from the Eikonal equation and we assume is only a function of x,then we find:

(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 Fermats principle, we can visualize that directionof rays follow the gradient of

phase front:

(D20)z-direction:

(D21)

x-direction: (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:


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. (D25)Where, n0is the index of refraction of water (1.34), is the index change insaturated 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:

2

42

2

421

2

Dt

x

Dt

xe Dt

x

(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.


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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:

(D33)If we assume C=, 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

refraction is varying in a centro-symmetric fashion. Thus we assume

is only a

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)

Index of

refraction

n(x)

dz

dx


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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, 7thed., p. 130, 157-158)

Example: Maxwells 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,

(D45)where indicates a launching angle at the initial point(r0, ).


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Observation:

- To obtain the rays of such a lens of Equation(D45) into Cartesian coordinates, we

use the transformation:

,

:

(D46) (D47)Equation (D47) clearly indicates each ray forms a circle with radius .

- 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 isachievable at the LHS side if we simply ask . Therefore, all rays leaving apoint P(r0, 0) will refocus back to a point Q , forming an inverted imagewith respect to P, and the magnification is

.

Figure 8. Ray Schematics of Maxwells Fish Eye Lenswith 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 .

a

(r0, 0).P

Q

r=a


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- 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 lens iiiwas 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.

Fermats Principle of least time

At first glance, Fermats 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 Lagrangianor Hamiltonianapproach in more detail in the coming lectures

aboutgradient index optics).


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The underlying argument is, light propagating between two given points Pand 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 vis 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 fromP

to P, and vice versa.

How is it consistent with wave picture? Modern theorists like Feynman take morerigorous 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

Fermats principle.

Endnotes and References

iFor reference about the high frequency (also coined as Physical Optics) limit of

Maxwell equations, you may read:

-Giorgio Franceschetti, Chapter 5: High Frequency Fields in Electromagnetics:

Theory, Techniques, and Engineering Paradigms,Plenum press, 1997.

iiSee 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 forsubwavelength imaging with positive refraction, New. J. Phys. 13, 033016 (2011).

- Juan C Gonzlez et al , Perfect drain for the Maxwell fish eye lens, New J. Phys. 13

023038 (2011).


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- R J Blaikie, Perfect imaging without refraction? New J. Phys. 13 125006 (2011).

- Xiang Zhang, "No drain, no gain"(Comment), Nature, 2011))

iiiReferences 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, 379381

(1974).

- S. K. Yao and D. B. Anderson, Shadow sputtered diffraction-limited waveguide

Luneburg lenses, Appl. Phys. Lett. 33, 307309 (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, 40674079 (1979).

- E. Colombini, Design of thin-film Luneburg lenses for maximum focal length

control, Appl. Opt. 20, 35893593 (1981).

- E. Colombini, Index-profile computation for the generalized Luneburg lens, J. Opt.

Soc. Am. 71, 14031405(1981).

Short Course Lecture 1 EM Waves and Eikonal Equations

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