Light Propagation (part 1)
Spatial Behavior

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Segment 2

2. Review and introduction

2.1 Newton’s corpuscular theory of light

Some recommended reading (besides Fowles): 
http://www.olympusmicro.com/primer/lightandcolor/particleorwave.html

Corpuscular vs wave: diffraction phenomenon

Wave diffraction

2.2 Huygens’ wave theory of light

Christiaan Huygens (1629 – 1695)

Huygens’ theory  (ca. 1678): 



Leading to the most important experimental result in early 19th century: interference (and implicitly diffraction) that confirmed the wave behavior of light.

For those interested, see more:

  
A History of Optics from Greek Antiquity to the Nineteenth Century
Olivier Darrigol, Oxford Univ. Press Inc., New York (2012).

Examples are all around us, in nature, e. g.

In the above fig. one should say “interference and diffraction”

2.3 Wave optics and ray optics (geometrical optics)

- Wave optics is the mathematicaly rigorous “classical” theory of light (which does not include quantum theory.
- However, wave optics alone is not sufficient to describe many other phenomena. The more complete theories would be - in this order: quantum electrodynamics (QED), quantum electro-weak theory, and GUT - grand -unifying theory,... We don’t really care for the latters, but quantum optics, a part of QED dealing with photons, the particle of light with low energy in the electron-Volt (eV) energy range is the basis for all optics phenomena of interest.
- Ray (or geometrical) optics is a mathematical approximation of wave optics when the wavelength is very small compared with the sizes of the objects or geometrical features of interest. Since this is often true in many optical system applications, it is not unusual to see that ray optics, which is intuitively easier to apply is often used instead of wave optics.

- This course will principally focus on wave optics, with the quantum theory of light is applied whenever it is relevant.

3. Maxwell's EM theory

Note: in the following, “gray background” parts are heavy math parts and you can “glaze” over if find them too difficult. Pay attention to those equations highlighted in green.

3.1 Basic review

Electromagnetic field is represented by 2 vectors: E: electric field and B: magnetic induction field. To describe the EM field in a medium, we need two more vector fields: electric dispacement field D, and magnetic vector field H. In addition, the sources of these fields are also needed in the Maxwell equations: J: electric current density, and ρ: the electric charge density.

Excersise: What are the units of all these fields and sources?

Wonder if there is "magnetic monopole” source ?

Maxwell equations:
∇×E+(1/c)((∂ B)/(∂ t)) = 0                            (3.1.1a)
∇×H-(1/c)((∂ D)/(∂ t)) =((4 π)/c) J                        (3.1.1b)
∇. D=4 π ρ                                    (3.1.1c)
∇. B=0                                         (3.1.1d)
The relations between various fields in a medium are:
D=E+4 π P                                   (3.1.2a)
B=H+4 π M                                  (3.1.2b)
J=σ . E                                          (3.1.2c)
where P is the polarization vector, M is the magnetization vector, and σ is the conductivity tensor. A most important relation for a material is:
 P = χ . E  ,                                   (3.1.3)
where χ is a second-rank tensor called electric susceptibility, and
  M = η . H                                     (3.1.4)
is the magnetization. In these cases, the materials are said to be linear, to reflect the linear relationship of various induced quantities with respect to the field.
In the simplest case, all material parameters are scalar (isotropic). For most materials:
B=μ H                                        (3.1.5)
where μ is just a scalar. Of course, there are interesting metamaterials but these will be discussed later in advanced topics (at the end of the course)
For nonlinear optics, the most interesting case is when χ is not a constant of E, but depending on E:  
 P = χ[E] . E  =  (χ^((1))) E + (χ^((2))) :E E+(χ^((3))) :E E E + ...                     (3.1.6)
where (χ^((n))) is called the nth order electric susceptibility tensor. Nonlinear optics is a crucial element of photonics.

ECE5358 - APP 1.3.1.1 - ELECTRON CLOUD POLARIZATION

Discussion: what are important devices, effects with optical nonlinearities?

The simplest case is, of course:  
D=E+4 π P =E+4 π  (χ^((1))) E =(1+4 π  (χ^((1)))) E  =  ε . E                   (3.1.7a)
B=H+4 π M =H+4 π η . H =  (1+4 π  η) H  =  μ . H                     (3.1.7b)
Then, by substituting:
                ∇×E+(1/c)((∂ B)/(∂ t)) = 0   => ∇×E+(1/c)(∂/(∂ t)) (H+4 π M)= 0             (3.1.8a)
Or:                            ∇×E=-(1/c)((∂ )/(∂ t)) (μ . H)=-(1/c)μ((∂ H)/(∂ t))                       (3.1.8b)
(notice the assumption of instantaneous time-response of M).
Likewise:
              ∇×H-(1/c)((∂ D)/(∂ t)) =((4 π)/c) J    => ∇×H-(1/c)(∂/(∂ t)) ( ε . E ) =((4 π)/c) (Subscript[σ, e]) . E      (3.1.9a)
                                    => ∇×H-(1/c) ε .((∂ E)/(∂ t)) =((4 π)/c) (Subscript[σ, e]) . E                   (3.1.9b)
And:                                     ∇. ϵ E=4 π ρ                                     (3.1.10a)
                                             ∇. μ H=0                                         (3.1.10b)
We now deal only with E and H in a linear medium.

As mentioned about  μ above, a key assumption we will making througout this course is that μ is isotropic and constant in space and instantaneous in time response. This is true about μ in all known practical optical materials around us: dielectric such as glasses (the types used in optical fibers and various optical devices), semiconductors, other optical dielectrics from inorganic to organic. In fact, we will often set μ =1. Of course, when we say “optical” here, we mean from long wave IR in the 10’s μm to blue/UV in the 10’s of nm. This is not true for longer waves and the response of  μ should be treated the same as that of ϵ for electric polarization.

We cannot say the same for ϵ being isotropic and constant in space and instantaneous in time response. A key property, we will see is that ϵ has a time-response (from the polarization field P) which makes the propagation of light in media qualitatively different from in vacuum.

3.2 Frequency domain

A very crucial concept is the nature of light when interacting with matters, i. e. atoms/molecules, more specifically, the electrons of atoms/molecules. As it turns out, as mentioned in 2.3 above, light is fundamentally made up of photons that are described with harmonic functions sine and cosine, or in complex notation, (e^(-i ω t)). (we use the convention of minus sign).

What it means is that we can assume that the time behavior of any light is made up of the (e^(-i ω t)) basis function:
                         E[(OverVector[r]);t]=∫a[(OverVector[r]);ω] (e^(-i ω t))dω     (3.2.1)
This is the essence of frequency domain and the basic principle of linear superposition.

Note: The minus sign associate with ω t is just a convention. For mathematical formalism, the integration extends from -∞ to ∞, but of course, there is no physical meaning of negative frequency.

Let’s assume for simplicity that we deal with E and B with only one frequency ω, called monochromatic field, then:

Maxwell equations:
∇×E+(1/c)((∂ B)/(∂ t)) = 0      (3.1.1a)   ⇒ ∇×E-i(ω/c)B = 0       (3.2.2a) 
∇×H-(1/c)((∂ D)/(∂ t)) =((4 π)/c) J   (3.1.1b)  ⇒ ∇×H+i(ω/c)D =((4 π)/c) J  (3.2.2b)
∇. D=4 π ρ     (3.1.1c)   remains the same
∇. B=0           (3.1.1d)  remains the same

We can still keep the relations between various fields in a medium:
D=E+4 π P                                   (3.1.2a)
B=H+4 π M                                  (3.1.2b)
J=σ . E                                          (3.1.2c)

with the knowledge that each quantity is a component with frequency ω. Hence, the key point here is that we have to be explicit about ω where it is relevant. Thus, for polarization  P and magnetization  M:
 P = χ[ω] . E  ,                                   (3.2.3)
where χ is a second-rank tensor called electric susceptibility, and
  M = η[ω] . H                                     (3.2.4)
In these cases, the materials are said to be linear, otherwise it would involve higher order harmonic components 2 ω, 3 ω, ...
In fact, for nonlinear optics, we have to include higher order (χ^((n))) :  
 P =   (χ^((1)))[ω] E + (χ^((2)))[2 ω=ω+ω]:E E+ (χ^((2)))[0=±ω∓ω]:E (SuperStar[E])
                                      +(χ^((3))) [3 ω=ω+ω+ω]:E E E + +(χ^((3))) [±ω=±(ω+ω)∓ω]:E E (SuperStar[E])...                     (3.2.5)

Each of the terms (χ^((n))) is not the same even of the same order n. For example, (χ^((2)))[2 ω=ω+ω] is the susceptibility that creates a polarization with double frequency of that of the incident E field (2 photons add up), where as (χ^((2)))[0=±ω∓ω] involves a photon cancels another in such a way that it gives a DC (zero frequency) polarization.

The essence is that time is explicitly removed and it becomes all-space differential equations.

The frequency domain is the essential approach to virtually all phenomena studied in this course.

That said, of course one can use time-domain. But this is mostly for numerical method only, not for analytical approach (time-domain would be too complex mathematically, except for few simplest problems). Example is FDTD. (link here). but you will see later that FDTD for optics is also quite involved with heavy number crunching. Thus, sometimes, frequency-domain approximation is more convenient with just slightly less accuracy.

3.3 Supplementary discussion on CGS-Gaussian and MKSA

3.3.1 Table of conversion

Useful table for CGS (Gaussian)-MKS conversion of equations and quantities  
({{Quantity, CGS-Gaussian, MKSA}, {Speed of light, c, (1/(Sqrt[(Subscript[ϵ, 0]) (Subscript[μ, 0])]))}, {E field/Potential/Voltage, E/Φ/V, (Sqrt[4 π (Subscript[ϵ, 0])])(E/Φ/V)}, {Displacement, D, (Sqrt[((4 π)/(Subscript[ϵ, 0])) ]) D}, {Charge density/Charge/
Current density/Current/Polarization, ρ/q/J/I/P, (1/(Sqrt[4 π (Subscript[ϵ, 0])]))(ρ/q/J/I/P)}, {Magnetic induction, B, (Sqrt[((4 π)/(Subscript[μ, 0])) ])B}, {Magnetic field, H, (Sqrt[4 π (Subscript[μ, 0])])H}, {Magnetization, M, (Sqrt[((Subscript[μ, 0])/(4 π)) ])M}, {Conductivity, σ, (σ/(4 π (Subscript[ϵ, 0])))}, {Dielectric constant, ϵ, (ϵ/(Subscript[ϵ, 0]))}, {Permeability, μ, (μ/(Subscript[μ, 0]))}, {Resistance/Impedance, R/Z, 4 π (Subscript[ϵ, 0]) (R/Z)}, {Inductance, L, 4 π (Subscript[ϵ, 0]) L}, {Capacitance, C, (C/(4 π (Subscript[ϵ, 0])))}})

3.3.2 MKSA Maxwell equations

The MKSA Maxwell equations are:
                                ∇×E+((∂ B)/(∂ t)) = 0                   
                                 ∇×H=((∂ D)/(∂ t)) + J   
                                  ∇.D= ρ 
                                  ∇.B=0  
and:                           D=(Subscript[ϵ, 0])E+P  
                                  (Sqrt[((4 π)/(Subscript[μ, 0])) ])B=(Sqrt[4 π (Subscript[μ, 0])])H+4 π (Sqrt[((Subscript[μ, 0])/(4 π)) ]) M  
                                  (Sqrt[(1/(Subscript[μ, 0])) ])B=(Sqrt[ (Subscript[μ, 0])])H+ (Sqrt[(Subscript[μ, 0]) ]) M  
                                 B=(Subscript[μ, 0]) (H+M)  
Thus:                       ∇×H=((∂ D)/(∂ t)) + J  =(Subscript[ϵ, 0]) (1+4 π χ)((∂ E)/(∂ t)) +σ E     
                               ∇×H=(Subscript[ϵ, 0]) ϵ((∂ E)/(∂ t)) +σ E     
and                          ∇×E+((∂ B)/(∂ t)) = 0 = ∇×E+((∂ )/(∂ t)) (Subscript[μ, 0]) (H+M)   
                                 0= ∇×E+((∂ )/(∂ t)) (Subscript[μ, 0]) (H+4 π η H)    
                                 0= ∇×E+(Subscript[μ, 0]) μ((∂ )/(∂ t))H  
In some case, people include the magnetization current:
                                        ∇×E=-(Subscript[μ, 0]) μ((∂ H)/(∂ t)) -(Subscript[σ, m]) H (with additional magnetization current)
                                         ∇×H=(Subscript[ϵ, 0])ϵ((∂ E)/(∂ t)) + (Subscript[σ, e]) . E  
The vacuum permittivity and permeability (Subscript[ϵ, 0]) and (Subscript[μ, 0]), respectively, are just scalar constants. Since this is included in software that is MKS based, we only have to input ϵ, μ, (Subscript[σ, m]), and (Subscript[σ, e]).

<<PhysicalConstants`

(Subscript[ϵ, 0])=VacuumPermittivity
(Subscript[μ, 0])=VacuumPermeability

Quantity["VacuumPermittivity"]

In this course, we will use the CGS-Gaussian system (except when we explicitly use MKSA for certain specific reason).

Assignment 1.3

4. The wave equation

4.1 The existence of EM waves

The biggest implication of the Maxwell equations is the existence of EM wave. 

∇×(∇×E+(1/c)((∂ B)/(∂ t)))= 0                           (4.1.1a)
∇(∇.E)-(∇^2)E +(1/c)((∂ )/(∂ t))∇×B= 0             (4.1.1.b)
Substitute 
∇×B = ( ∇×µ H )=µ (((4 π)/c) J +(1/c)((∂ D)/(∂ t)))
                                       = µ (((4 π)/c) J +(1/c)((∂ )/(∂ t))(E+ 4 π P))     (4.1.2)
and:
∇.E+  4 π ∇. P= ((4 π)/c) ρ                      (4.1.3)
we obtain:
(∇^2)E -((4 π)/c) ∇(ρ -c ∇. P)-(1/c)((∂ )/(∂ t))∇×B= 0                                        (4.1.4a)
(∇^2)E -((4 π)/c) ∇(ρ -c ∇. P)-(1/c)((∂ )/(∂ t))µ (((4 π)/c) J +(1/c)((∂ )/(∂ t))(E+ 4 π P))= 0  (4.1.4b)
(∇^2)E -(µ/(c^2))((∂^2)/(∂ (t^2)))(E + 4 π P) = ((4 π)/c) ∇(ρ -c ∇. P)+((4 π)/(c^2))µ((∂ )/(∂ t)) J       (4.1.4c)

The above equations are still fundamental, although they are the derived, not original Maxwell’s equations. We have not made any particular assumption yet about P and ϵ  except for the one about µ being isotropic and constant vs. time. 

In the absence of sources (for macroscopic classical optics, where quantum interaction of the photon and matter is statistically averaged), both ρ and J are zero (not usually the case for RF-mirowave).

Let   P =  (χ^((1))) E + ℙ                                                  (4.1.5)     
where    ℙ  denotes any residual polarization besides the linear polarization component   (χ^((1))) E , which is the non-linear component of polarization that is important for non-linear optics. Then:
   
E + 4 π P= (1+4 π (χ^((1)))) E + 4 π ℙ = ϵ.E +  4 π ℙ           (4.1.6)
∇. P =  ∇. (χ^((1))) E + ∇.ℙ.                                                    (4.1.7)
The equation (4.1.4c):                             
(∇^2)E -(µ/(c^2))((∂^2)/(∂ (t^2)))(E + 4 π P) = ((4 π)/c) ∇(ρ -c ∇. P)+((4 π)/(c^2))µ((∂ )/(∂ t)) J    
becomes:
(∇^2)E -(μ/(c^2))((∂^2)/(∂ (t^2))) ϵ.E =((4 π)/(c^2))((∂^2)/(∂ (t^2))) ℙ  - 4 π ∇ ( ∇. (χ^((1))) E + ∇.ℙ )    (4.1.8a)
(∇^2)E + 4 π ∇ ( ∇. (χ^((1)))E ) -(μ/(c^2))((∂^2)/(∂ (t^2))) ϵ.E =((4 π)/(c^2))((∂^2)/(∂ (t^2))) ℙ  - 4 π  ∇ (∇.ℙ)   (4.1.8b)
This equation is essential for dealing with nonvanishing  ℙ in the case of nonlinear optics. 

When there is no nonlinearity, i. e.     ℙ =0 
(∇^2)E + 4 π ∇ ( ∇. (χ^((1)))E ) -(μ/(c^2))((∂^2)/(∂ (t^2))) ϵ.E =0               (4.1.9)
In an anisotropic medium, ϵ, (χ^((1))) are tensor and the above equation must be solved simultaneously with E and D. Again, if the medium is isotropic, uniform,  ∇. E = (1/ϵ)  ∇. D = 0, and:

(∇^2)E  -((μ ϵ)/(c^2))((∂^2)/(∂ (t^2)))E =0                              (4.1.10)
This is the wave equation. In vacuum, μ= 1, ϵ=1 and the wave propagates with the speed of light. We have a similar equation for H:
(∇^2)H  -((μ ϵ)/(c^2))((∂^2)/(∂ (t^2)))H =0                             (4.1.11)

Remember: The above equations are correct for vacuum, but not necessarily in a medium. It is approximately correct so far only when the assumptions in the above are true for P and χ. We will see that we don’t really use these equations often, but the more relevant Helmholtz equation below. However, they are useful to illustrate the nature of EM waves.

Remember that polarization field is a response of the medium to the EM field. Hence, one should think causally that:
                                      P[t]=(∫_(-∞)^t)α[t-τ] E[τ]dτ         (4.1.12)
In other words, the polarization at time t does not just depend on the instantaneous E field at time t, but also on the history of the field before that time because the medium response is an accumulation (integral) effect of the atomic/molecular eletron response to the entire history of the field. (For simplicity, ignore spatial neighboring effect). It appears that one can treat the problem as a linear time-invariant system and can use the time-domain approach.

For a perfect LTI system, the frequency response is the FT of the impulse response α[t]: (causally, α[t]=0 for t<0).
                                           χ[ω]=∫α[t](e^(i ω t)) dt         (4.1.13)
But as we discuss in 3.2, the truly correct approach for analytical perspective or even also numerical approximation is the frequency-domain response approach, which is consistent with the quantum theory of the interaction of light and matter. Hence, instead of dealing with (4.1.12), we can write:
                                     P[ω]=χ[ω] E[ω]          (4.1.14)
We will also see that the Helmholtz equation is the convenient starting point to deal with waves in media.

4.2 Vector potential

In a uniform, linear, homogeneous medium, the Maxwell Eqs. can be expressed in simpler terms: 
∇×E+(1/c)((∂ B)/(∂ t)) = 0                             (4.2.1a)
∇×B-((µ ε)/c)((∂ E)/(∂ t)) =((4 π)/c)µ J                     (4.2.1b)
∇.E=((4 π)/(c ε)) ρ                                        (4.2.1c)
∇.B=0                                              (4.2.1d)
We can obtain the same wave eqs. as we did earlier. But does that mean we have to solve for all six components: 3 for E and 3 for B? Obviously not, because they are related (dependent on each other) as shown. The question is: what are the minimum components we should solve? That certainly depends on specific problem. But one very useful insight is the concept of vector potential.

Since ∇.B=0 , and div of any curl is zero, we postulate the existence of a vector field such that:              B=∇×A.                   (4.2.2)
Then:
∇×(E+(1/c)((∂ A)/(∂ t)))= 0 .                        (4.2.3)
Since Curl of any grad is also zero, we can also postulate:
E+(1/c)((∂ A)/(∂ t))=-∇ϕ .                            (4.2.4)
Then the eqs. become:
∇×B-((µ ε)/c)((∂ E)/(∂ t)) =((4 π)/c)µ J   ==> ∇×(∇×A)-((µ ε)/c)((∂ )/(∂ t))(-(1/c)((∂ A)/(∂ t))-∇ϕ ) =((4 π)/c)µ J  (4.2.5)
or: 
∇(∇.A)-(∇^2)A+((µ ε)/(c^2))(((∂^2) A)/(∂ (t^2))) =((4 π)/c)µ J -((µ ε)/c)∇((∂ ϕ)/(∂ t))         (4.2.6)
(∇^2)A-((µ ε)/(c^2))(((∂^2) A)/(∂ (t^2))) =-((4 π)/c) µ J +∇(∇.A+((µ ε)/c)((∂ ϕ)/(∂ t)))         (4.2.7)
Likewise: 
∇.E=((4 π)/(c ε)) ρ ==>   (1/c)∇.((∂ A)/(∂ t))+(∇^2)ϕ =-((4 π)/(c ε)) ρ .           (4.2.8)
or: (∇^2)ϕ- ((µ ε)/(c^2))(((∂^2) ϕ)/(∂ (t^2)))=-((4 π)/(c ε)) ρ -(1/c)((∂ )/(∂ t))(∇.A+ ((µ ε)/c)((∂ ϕ)/(∂ t))) .  (4.2.9)

If we impose the Lorentz condition: ∇.A+ ((µ ε)/c)((∂ ϕ)/(∂ t))=0  which connects the two potential (so that they are not independent anymore), we obtain homogeneous eqs. :  
(∇^2)A-((µ ε)/(c^2))(((∂^2) A)/(∂ (t^2))) =-((4 π)/c) µ J                    (4.2.10a)
(∇^2)ϕ- ((µ ε)/(c^2))(((∂^2) ϕ)/(∂ (t^2)))=-((4 π)/(c ε)) ρ                        (4.2.10b)
Now we see that we can solve for A, ϕ and we obtain E and B. However, it can seen that there is not a unique A, ϕ: if we define new A' and ϕ' such that:
A'=A+∇χ                                       (4.2.11a)
ϕ'=ϕ-(1/c)((∂ χ)/(∂ t))                                     (4.2.11b)
where χ is any arbitrary scalar function,
then:                    ∇×A'=∇×A=B;                       (4.2.12a)
and
       E=-(1/c)((∂ A')/(∂ t))-∇ϕ'=-(1/c)((∂ A)/(∂ t))-(1/c)((∂ )/(∂ t))∇χ-∇ϕ+(1/c)∇((∂ χ)/(∂ t))
   =-(1/c)((∂ A)/(∂ t))-∇ϕ   (4.2.12b)
which yield the same E and B. With Lorentz condition:
(∇^2)χ- ((µ ε)/(c^2))(((∂^2)χ)/(∂ (t^2)))=  0,                           (4.2.13)
which satisfies the homogeneous wave equation also. This transformation is known as "gauge transformation".
A special case when there is no charge, one can choose χ to be c∫ϕd t since this function satifies the wave equation, then: ϕ'=0 and it is sufficient to solve only for A, from which we obtain: 
E=-(1/c)((∂ A)/(∂ t)) ; B=∇×A; and Lorentz condition ∇.A        (4.2.14)

The key point here is that we see that at most 4 components, A and ϕ are sufficient to determine E and H (but A and ϕ are also dependent on each other with the Lorentz condition). We don’t have truly 6 independent components of E and H. If they were, there wouldn’t be any Maxwell’s equation between them! and there is no theory of light. In many problems, the number of independent component can even be reduce to as low as 1. In other words, in some problem, we need to determine only 1 component and can derive all 6 components of E and H. The more symmetry a problem has, the fewer is the number of components. A and ϕ are the starting concept for quantum field theory, where the photons are described with such a field, not E and H.

Summary:
1- Maxwell equations are deceptively simple. Rarely do we use them in the original form as the starting point to solve a problem. We use derived forms (secondary equations) to reduce the coupling of various terms.  
2- The secondary (derived) forms can be very complicated or simple depending on the problems.
3- It is important to know which secondary form is suitable for which case: it is crucial to understand (and remember or check if you don't remember) the underlying assumptions that give a specific form. A very common error in setting up a problem is to use an equation whose assumption is not valid for the case.