Lecture 39

Quantum Coherent State of Light

39.1 Quantum Coherent State of Light

We have seen that a photon number states1 of a quantum pendulum do not have a classical correspondence as the average or expectation values of the position and momentum of the pendulum are always zero for all time for this state. Therefore, we have to seek a time-dependent quantum state that has the classical equivalence of a pendulum. This is the coherent state, which is the contribution of many researchers, most notably, George Sudarshan (1931–2018) [237] and Roy Glauber (1925–2018) [238] in 1963. Glauber was awarded the Nobel prize in 2005.

We like to emphasize again that the modes of an electromagnetic cavity oscillation are homomorphic to the oscillation of classical pendulum. Hence, we first connect the oscillation of a quantum pendulum to a classical pendulum. Then we can connect the oscillation of a quantum electromagnetic mode to the classical electromagnetic mode and then to the quantum pendulum.

39.1.1 Quantum Harmonic Oscillator Revisited

To this end, we revisit the quantum harmonic oscillator or the quantum pendulum with more mathematical depth. Rewriting Schrödinger equation as the eigenequation for the photon number state for the quantum harmonic oscillator, we have

Hψˆ

n(x) = [

− ~2

2m

d2

dx2 +

1

2mω2

0x2] ψn(x) = Enψn(x). (39.1.1)

where ψn(x) is the eigenfunction, and En is the eigenvalue. The above can be changed into a dimensionless form first by dividing ~ω0, and then let ξ =

√mω0

~

x be a dimensionless

1 In quantum theory, a “state” is synonymous with a state vector or a function.

401

402 Electromagnetic Field Theory

variable. The above then becomes

1

2

(− d2

dξ2 + ξ2) ψ(ξ) = E

~ω0

ψ(ξ) (39.1.2)

We can define πˆ = −i d

dξ and ξˆ = Iˆξ to rewrite the Hamiltonian as

Hˆ = 1

2~ω0(πˆ2 + ξˆ2) (39.1.3)

Furthermore, the Hamiltonian in (39.1.2) looks almost like A2 − B2, and hence motivates its factorization. To this end, we first show that

1√2

(− d

dξ + ξ)

1√2

(d

dξ + ξ)

=

1

2

(− d2

dξ2 + ξ2) −

1

2

(d

dξ ξ − ξ d

dξ) (39.1.4)

It can be shown easily that as operators (meaning that they will act on a function to their right), the last term on the right-hand side is an identity operator, namely that

(d

dξ ξ − ξ d

dξ) = Iˆ (39.1.5)

Therefore

1

2

(− d2

dξ2 + ξ2) =

1√2

(− d

dξ + ξ)

1√2

(d

dξ + ξ) +

1

2

(39.1.6)

We define the operator

aˆ† =

1√2

(− d

dξ + ξ) (39.1.7)

The above is the creations, or raising operator and the reason for its name is obviated later. Moreover, we define

aˆ =

1√2

(d

dξ + ξ) (39.1.8)

which represents the annihilation or lowering operator. With the above definitions of the raising and lowering operators, it is easy to show that by straightforward substitution that

[a, ˆ aˆ†] = aˆaˆ† − aˆ†aˆ = Iˆ (39.1.9)

Therefore, Schrödinger equation (39.1.2) for quantum harmonic oscillator can be rewritten more concisely as

1

2

(aˆ†aˆ + aˆaˆ†) ψ =

(aˆ†aˆ +

1

2) ψ =

E

~ω0

ψ (39.1.10)

403 Quantum Coherent State of Light

In mathematics, a function is analogous to a vector. So ψ is the implicit representation of a vector. The operator

(aˆ†aˆ +

1

2)

is an implicit2 representation of an operator, and in this case, a differential operator. So in the above, (39.1.10), is analogous to the matrix eigenvalue equation A · x = λx.

Consequently, the Hamiltonian operator can be expressed concisely as

Hˆ = ~ω0(aˆ†aˆ +

1

2) (39.1.11)

Equation (39.1.10) above is in implicit math notation. In implicit Dirac notation, it is

(aˆ†aˆ +

1

2) |ψi =

E

~ω0

|ψi (39.1.12)

In the above, ψ(ξ) is a function which is a vector in a functional space. It is denoted as ψ in math notation and |ψi in Dirac notation. This is also known as the “ket”. The conjugate transpose of a vector in Dirac notation is called a “bra” which is denoted as hψ|. Hence, the inner product between two vectors is denoted as hψ1|ψ2i in Dirac notation.3

If we denote a photon number state by ψn(x) in explicit notation, ψn in math notation or |ψni in Dirac notation, then we have

(aˆ†aˆ +

1

2) |ψni =

En

~ω0

|ψni =

(n +

1

2) |ψni (39.1.13)

where we have used the fact that En = (n + 1/2)~ω0. Therefore, by comparing terms in the above, we have

aˆ†aˆ|ψni = n|ψni (39.1.14)

and the operator aˆ†aˆ is also known as the number operator because of the above. It is often denoted as

nˆ = aˆ†aˆ (39.1.15)

and |ψni is an eigenvector of nˆ = aˆ†aˆ operator with eigenvalue n. It can be further shown by direct substitution that

aˆ|ψni =

√n|ψn−1i ⇔ aˆ|ni =

√n|n − 1i (39.1.16)

aˆ†|ψni =

√n + 1|ψn+1i ⇔ aˆ†|ni =

√n + 1|n + 1i (39.1.17)

hence their names as lowering and raising operator.4

2 A notation like A · x, we will call implicit, while a notation Pi,j Aijxj, we will call explicit.

3 There is a one-to-one correspondence of Dirac notation to matrix algebra notation. Aˆ|xi ↔ A·x, hx| ↔ x† hx1|x2i ↔ x†1 · x2.

4 The above notation for a vector could appear cryptic or too terse to the uninitiated. To parse it, one can always down-convert from an abstract notation to a more explicit notation. Namely, |ni → |ψni → ψn(ξ).

404 Electromagnetic Field Theory

39.2 Some Words on Quantum Randomness and Quantum Observables

We saw previously that in classical mechanics, the conjugate variables p and x are deterministic variables. But in the quantum world, they become random variables with means and variance. It was quite easy to see that x is a random variable in the quantum world. But the momentum p is elevated to become a differential operator pˆ, and it is not clear that it is a random variable anymore.

Quantum theory is a lot richer in content than classical theory. Hence, in quantum theory, conjugate variables like p and x are observables endowed with the properties of mean and variance. For them to be endowed with these properties, they are elevated to become quantum operators, which are the representations of these observables. To be meaningful, a quantum state |ψi has to be defined for a quantum system, and these operators represent observables act on the quantum state.

Henceforth, we have to extend the concept of the average of a random variable to the “average” of a quantum operator. Now that we know Dirac notation, we can write the expectation value of the operator pˆ with respect to a quantum state ψ as

hpˆi = hψ|pˆ|ψi = p¯ (39.2.1)

The above is the elevated way of taking the “average” of an operator which is related to the mean of the random variable p.

As mentioned before, Dirac notation is homomorphic to matrix algebra notation. The above is similar to ψ† · P · ψ = p¯. This quantity p¯ is always real if P is a Hermitian matrix. Hence, in (39.2.1), the expectation value p¯ is always real if pˆ is Hermitian. In fact, it can be proved that pˆ is Hermitian in the function space that it is defined.

Furthermore, the variance of the random variable p can be derived from the quantum operator pˆ with respect to to a quantum state |ψi. It is defined as

σp2 = hpˆ2i − hpˆi2 (39.2.2)

where σp is the standard deviation of the random variable p and σp2 is its variance.

The above implies that the definition of the quantum operators and the quantum states is not unique. One can define a unitary matrix or operator U such that U† · U = I. Then the new quantum state is now given by the unitary transform ψ0 = U · ψ. With this, we can easily show that

p¯ = ψ† · P · ψ = ψ† · U† · U · P · U† · U · ψ

= ψ0† · P0 · ψ0 (39.2.3)

where P0 = U·P·U† via unitary transform. Now, P0 is the new quantum operator representing the observable p and ψ0 is the new quantum state vector.

In the previous section, we have elevated the position variable ξ to become an operator ξˆ = ξ Iˆ. This operator is clearly Hermitian, and hence, the expectation value of this position operator is always real. Here, ξˆ is diagonal in the coordinate representation, but it need not be in other representations.

Quantum Coherent State of Light 405

39.3 Derivation of the Coherent States

As one cannot see the characteristics of a classical pendulum emerging from the photon number states, one needs another way of bridging the quantum world with the classical world. This is the role of the coherent state: It will show the correspondence principle, with a classical pendulum emerging from a quantum pendulum when the energy of the pendulum is large. Hence, it will be interesting to see how the coherent state is derived. The derivation of the coherent state is more math than physics. Nevertheless, the derivation is interesting. We are going to present it according to the simplest way presented in the literature. There are deeper mathematical methods to derive this coherent state like Bogoliubov transform which is outside the scope of this course.

Now, endowed with the needed mathematical tools, we can derive the coherent state. To say succinctly, the coherent state is the eigenstate of the annihilation operator, namely that

aˆ|αi = α|αi (39.3.1)

Here, we use α as an eigenvalue as well as an index or identifier of the state |αi.5 Since the number state |ni is complete, the coherent state |αi can be expanded in terms of the number state |ni. Or that

|αi =

Σ∞

n=0 Cn|ni (39.3.2)

When the annihilation operator is applied to the above, we have

aˆ|αi =

Σ∞

n=0 Cnaˆ|ni =

Σ∞

n=1 Cnaˆ|ni =

Σ∞

n=1 Cn

√n|n − 1i

=

Σ∞

n=0 Cn+1

√n + 1|ni (39.3.3)

Equating the above with α|αi, then

Σ∞

n=0 Cn+1

√n + 1|ni = α

Σ∞

n=0 Cn|ni (39.3.4)

By the orthonormality of the number states |ni, then we can take the inner product of the above with hn| and making use of the orthonormal relation that hn0|ni = δn0n to remove the summation sign. Then we arrive at

Cn+1 = αCn/√n + 1 (39.3.5)

Or recursively

Cn = Cn−1α/√n = Cn−2α2/√n(n − 1) = . . . = C0αn/√n! (39.3.6)

5 This notation is cryptic and terse, but one can always down-convert it as |αi → |fαi → fα(ξ) to get a more explicit notation.

406 Electromagnetic Field Theory

Consequently, the coherent state |αi is

|αi = C0

Σ∞

n=0 αn

√n!

|ni (39.3.7)

But due to the probabilistic interpretation of quantum mechanics, the state vector |αi is normalized to one, or that6

hα|αi = 1 (39.3.8)

Then

hα|αi = C0* C0

Σ∞

n,n0

αn

√n!

αn0

√n0!

hn0|ni

= |C0|2 Σ∞

n=0

|α|2n

n!

= |C0|2 e|α|2

= 1 (39.3.9)

Therefore, C0 = e−|α|2/2, or that

|αi = e−|α|2/2 Σ∞

n=0

αn

√n!

|ni (39.3.10)

In the above, to reduce the double summations into a single summation, we have make use of hn0|ni = δn0n, or that the photon-number states are orthonormal. Also since aˆ is not a Hermitian operator, its eigenvalue α can be a complex number.

Since the coherent state is a linear superposition of the photon number states, an average number of photons can be associated with the coherent state. If the average number of photons embedded in a coherent is N, then it can be shown that N = |α|2. As shall be shown, α is related to the amplitude of the quantum oscillation: The more photons there are, the larger is |α|.

39.3.1 Time Evolution of a Quantum State

The Schrodinger equation can be written concisely as

Hˆ |ψi = i~∂t|ψi (39.3.11)

The above not only entails the form of Schrodinger equation, it is the form of the general quantum state equation. Since Hˆ is time independent, the formal solution to the above is

|ψ(t)i = e−iHt/ˆ ~ |ψ(0)i (39.3.12)

Applying this to the photon number state with Hˆ being that of the quantum pendulum, then

e−iHt/ˆ ~ |ni = e−iωnt |ni (39.3.13)

6 The expression can be written more explicitly as hα|αi = hfα|fαi =

∫∞

∞ dξf∗α(ξ)fα(ξ) = 1.

407 Quantum Coherent State of Light

where ωn = (n + 1

2)ω0. The above simplification follows because |ni an eigenstate of the Hamiltonian Hˆ for the quantum pendulum. The above follows because

Hˆ |ni = ~ωn|ni = ~ω0(n +

1

2) |ni (39.3.14)

In other words, |ni is an eigenvector of Hˆ .

Time Evolution of the Coherent State

Using the above time-evolution operator, then the time dependent coherent state evolves in time as7

|α, ti = e−iHt/ˆ ~ |αi = e−|α|2/2 Σ∞

n=0 αn e−iωnt

√n!

|ni (39.3.15)

By letting ωn = ω0(n +

1

2), the above can be written as

|α, ti = e−iω0t/2 e−|α|2/2 Σ∞

n=0

(αe−iω0t)n

√n!

|ni (39.3.16)

= e−iω0t/2

|αe−iω0t i = e−iω0t/2 |α˜i (39.3.17)

where α˜ = αe−iω0t. Now we see that the last factor in (39.3.16) is similar to the expression for a coherent state in (39.3.10). Therefore, we can express the above more succinctly by replacing α in (39.3.10) with α˜ = αe−iω0t as

aˆ|α, ti = e−iω0t/2

(αe−iω0t) |αe−iω0t i = α˜|α, ti (39.3.18)

Therefore, |α, ti is the eigenfunction of the aˆ operator. But now, the eigenvalue of the annihilation operator aˆ is a complex number which is a function of time t.

39.4 More on the Creation and Annihilation Operator

In order to connect the quantum pendulum to a classical pendulum via the coherent state, we will introduce some new operators. Since

aˆ† =

1√2

(− d

dξ + ξ) (39.4.1)

aˆ =

1√2

(d

dξ + ξ) (39.4.2)

7 Note that |α, ti is a shorthand for fα(ξ, t).

408 Electromagnetic Field Theory

We can relate aˆ† and aˆ, which are non-hermitian, to the momentum operator πˆ and position operator ξˆ previously defined which are hermitian. Then

aˆ† =

1√2(−iπˆ + ξˆ) (39.4.3)

aˆ =

1√2(iπˆ + ξˆ) (39.4.4)

We also notice that

ξˆ =

1√2(aˆ† + aˆ) = ξ Iˆ (39.4.5)

πˆ =

i

√2(aˆ† − aˆ) = −i d

dξ (39.4.6)

Notice that both ξˆ and πˆ are Hermitian operators in the above, with real expectation values. With this, the average or expectation value of the position of the pendulum in normalized coordinate, ξ, can be found by taking expectation with respect to the coherent state, or

hα|ξˆ|αi =

1

√2 hα|aˆ† + aˆ|αi (39.4.7)

Since by taking the complex conjugation transpose of (39.3.1)8

hα|aˆ† = hα|α* (39.4.8)

and (39.4.7) becomes

ξ¯ = hξˆi = hα|ξˆ|αi =

1

√2(α* + α)hα|αi =

√2<e[α] 6= 0 (39.4.9)

Repeating the exercise for time-dependent case, when we let α → α˜(t) = αe−iω0t, then, letting α = |α|e−iψ, then

hξˆ(t)i =

√2|α| cos(ω0t + ψ) (39.4.10)

By the same token,

π¯ = hπˆi = hα|πˆ|αi =

i

√2(α* − α)hα|αi =

√2=m[α] 6= 0 (39.4.11)

For the time-dependent case, we let α → α˜(t) = αe−iω0t,

hπˆ(t)i = −√2|α|sin(ω0t + ψ) (39.4.12)

Hence, we see that the expectation values of the normalized coordinate and momentum just behave like a classical pendulum. There is however a marked difference: These values have

standard deviations or variances that are non-zero. Thus, they have quantum fluctuation or quantum noise associated with them. Since the quantum pendulum is homomorphic with the oscillation of a quantum electromagnetic mode, the amplitude of a quantum electromagnetic mode will have a mean and a fluctuation as well.

Figure 39.1: The time evolution of the coherent state. It follows the motion of a classical pendulum or harmonic oscillator (courtesy of Gerry and Knight [239]).

Figure 39.2: The time evolution of the coherent state for different α’s. The left figure is for α = 5 while the right figure is for α = 10. Recall that N = |α|2.

409 Quantum Coherent State of Light

39.4.1 Connecting Quantum Pendulum to Electromagnetic Oscillator

We see that the electromagnetic oscillator in a cavity is similar or homomorphic to a pendulum. To make the connection, we next have to elevate a classical pendulum to become a quantum pendulum. The classical Hamiltonian is

H = T + V =

p2

2m +

1

2mω2

0x2 =

1

2(P2(t) + Q2(t)) = E (39.4.13)

In the above, P is a normalized momentum and Q is a normalized coordinate, and their squares have the unit of energy. We have also shown that when the classical pendulum is elevated to be a quantum pendulum, then Schrödinger equation becomes

~ωl(aˆ†aˆ +

1

2) |ψ, ti = i~∂t|ψ, ti (39.4.14)

Our next task is to connect the electromagnetic oscillator to this pendulum. In general, the total energy or the Hamiltonian of an electromagnetic system is

H =

1

2

∫V dr

[εE2(r, t) + 1

µB2(r, t)] . (39.4.15)

It is customary to write this Hamiltonian in terms of scalar and vector potentials. For simplicity, we use a 1D cavity, and let A = xˆAx, ∇ · A = 0 so that ∂xAx = 0, and letting Φ = 0. Then B = ∇ × A and E = −.A, and the classical Hamiltonian from (39.4.15) for a Maxwellian system becomes

H =

1

2

∫V dr

[ε .A2(r, t) + 1

µ(∇ × A(r, t))2] . (39.4.16)

For the 1D case, the above implies that By = ∂zAx, and Ex = −∂tAx = −.Ax. Hence, we let

Ax = A0(t) sin(klz) (39.4.17)

Ex = −.A0(t) sin(klz) = E0(t) sin(klz) (39.4.18)

By = klA0(t) cos(klz). (39.4.19)

where E0(t) = −.A0(t). After integrating over the volume such that ∫V dr = A∫L0 dz, the Hamiltonian (39.4.16) then becomes

H =

V0ε

4

(.A0(t))2 +

V0

4µ k2l A02(t). (39.4.20)

where V0 = AL, is the mode volume. The form of (39.4.20) now resembles the pendulum Hamiltonian. We can think of A0(t) as being related to the displacement of the pendulum.

410 Electromagnetic Field Theory

Hence, the second term resembles the potential energy. The first term has the time derivative of A0(t), and hence, can be connected to the kinetic energy of the system. Therefore, we can rewrite the Hamiltonian as

H =

1

2(P2(t) + Q2(t)) (39.4.21)

where

P(t) = rV0ε

2

.A0(t) = −rV0ε

2

E0(t), Q(t) = sV0

2µ klA0(t) (39.4.22)

By elevating P and Q to be quantum operators,

P(t) → Pˆ =

√~ωlπˆ(t), Q(t) → Qˆ =

√~ωl ξˆ(t) (39.4.23)

so that the quantum Hamiltonian now is

Hˆ = 1

2[Pˆ2 + Qˆ2] = 1

2~ωl(πˆ2 + ξˆ2) (39.4.24)

similar to (39.1.3) as before except now that the resonant frequency of this mode is ωl instead of ω0 because these are the cavity modes, each of which is homomorphic to a quantum pendulum. An equation of motion for the state of the quantum system can be associated with the quantum Hamiltonian just as in the quantum pendulum case.

We have shown previously that

aˆ† + aˆ = √2 ξˆ (39.4.25)

aˆ† − aˆ = −√2iπˆ (39.4.26)

Then we can let

Pˆ = −rV0ε

2 Eˆ0 = √~ωlπˆ (39.4.27)

Finally, we arrive at

Eˆ0 = −r2~ωl

εV0 πˆ =

1

i r~ωl

εV0(aˆ† − aˆ) (39.4.28)

Now that E0 has been elevated to be a quantum operator Eˆ0, from (39.4.18), we can put in the space dependence to get

Eˆx(z) = Eˆ0 sin(klz) (39.4.29)

Consequently,

Eˆx(z) = 1

i r~ωl

εV0(aˆ† − aˆ) sin(klz) (39.4.30)

Notice that in the above, Eˆ0, and Eˆx(z) are all Hermitian operators and they correspond to quantum observables that have randomness associated with them. Also, the operators

are independent of time because they are in the Schrodinger picture. The derivation in the Heisenberg picture can be repeated.

In the Schrodinger picture, to get time dependence fields, one has to take the expectation value of the operators with respect to time-varying quantum state vector like the time-varying coherent state.

To let Eˆx have any meaning, it should act on a quantum state. For example,

|ψEi = Eˆx|ψi (39.4.31)

Notice that thus far, all the operators derived are independent of time. To derive time dependence of these operators, one needs to find their expectation value with respect to time-dependent state vectors.9

To illustrate this, we can take expectation value of the quantum operator Eˆx(z) with respect to a time dependent state vector, like the time-dependent coherent state, Thus

hEx(z, t)i = hα, t|Eˆx(z)|α, ti =

1

i r~ωl

εV0 hα, t|aˆ† − aˆ|α, ti

=

1

i r~ωl

εV0(α˜* (t) − α˜(t))hα, t|α, ti = −2 r~ωl

εV0 =m(α˜) (39.4.32)

Using the time-dependent α˜(t) = αe−iωlt = |α|e−i(ωlt+ψ) in the above, we have

hEx(z, t)i = 2r~ωl

εV0 |α|sin(ωlt + ψ) (39.4.33)

where α˜(t) = αe−iωlt. The expectation value of the operator with respect to a time-varying quantum state in fact gives rise to a time-varying quantity. The above, which is the average of a random field, resembles a classical field. But since it is rooted in a random variable, it has a standard deviation in addition to having a mean.

We can also show that

Bˆy(z) = klAˆ0 cos(klz) = r2µ~ωl

V0 ξˆ =

rµ~ωl

V0 (aˆ† + aˆ) (39.4.34)

Again, these are time-independent operators in the Schrodinger picture. To get time-dependent quantities, we have to take the expectation value of the above operator with respect to to a time-varying quantum state.

9 This is known as the Schrodinger picture.

Bibliography

[1] J. A. Kong, Theory of electromagnetic waves. New York, Wiley-Interscience, 1975.

[2] A. Einstein et al., “On the electrodynamics of moving bodies,” Annalen der Physik, vol. 17, no. 891, p. 50, 1905.

[3] P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 114, no. 767, pp. 243–265, 1927.

[4] R. J. Glauber, “Coherent and incoherent states of the radiation field,” Physical Review, vol. 131, no. 6, p. 2766, 1963.

[5] C.-N. Yang and R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Physical review, vol. 96, no. 1, p. 191, 1954.

[6] G. t’Hooft, 50 years of Yang-Mills theory. World Scientific, 2005.

[7] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Princeton University Press, 2017.

[8] F. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” Journal of Electromagnetic Waves and Applications, vol. 13, no. 5, pp. 665–686, 1999.

[9] W. C. Chew, E. Michielssen, J.-M. Jin, and J. Song, Fast and efficient algorithms in computational electromagnetics. Artech House, Inc., 2001.

[10] A. Volta, “On the electricity excited by the mere contact of conducting substances of different kinds. in a letter from Mr. Alexander Volta, FRS Professor of Natural Philosophy in the University of Pavia, to the Rt. Hon. Sir Joseph Banks, Bart. KBPR S,” Philosophical transactions of the Royal Society of London, no. 90, pp. 403–431, 1800.

[11] A.-M. Ampère, Exposé méthodique des phénomènes électro-dynamiques, et des lois de ces phénomènes. Bachelier, 1823.

[12] ——, Mémoire sur la théorie mathématique des phénomènes électro-dynamiques uniquement déduite de l’expérience: dans lequel se trouvent réunis les Mémoires que M. Ampère a communiqués à l’Académie royale des Sciences, dans les séances des 4 et

414 Electromagnetic Field Theory

26 décembre 1820, 10 juin 1822, 22 décembre 1823, 12 septembre et 21 novembre 1825. Bachelier, 1825.

[13] B. Jones and M. Faraday, The life and letters of Faraday. Cambridge University Press, 2010, vol. 2.

[14] G. Kirchhoff, “Ueber die auflösung der gleichungen, auf welche man bei der untersuchung der linearen vertheilung galvanischer ströme geführt wird,” Annalen der Physik, vol. 148, no. 12, pp. 497–508, 1847.

[15] L. Weinberg, “Kirchhoff’s’ third and fourth laws’,” IRE Transactions on Circuit Theory, vol. 5, no. 1, pp. 8–30, 1958.

[16] T. Standage, The Victorian Internet: The remarkable story of the telegraph and the nineteenth century’s online pioneers. Phoenix, 1998.

[17] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Philosophical transactions of the Royal Society of London, no. 155, pp. 459–512, 1865.

[18] H. Hertz, “On the finite velocity of propagation of electromagnetic actions,” Electric Waves, vol. 110, 1888.

[19] M. Romer and I. B. Cohen, “Roemer and the first determination of the velocity of light (1676),” Isis, vol. 31, no. 2, pp. 327–379, 1940.

[20] A. Arons and M. Peppard, “Einstein’s proposal of the photon concept–a translation of the Annalen der Physik paper of 1905,” American Journal of Physics, vol. 33, no. 5, pp. 367–374, 1965.

[21] A. Pais, “Einstein and the quantum theory,” Reviews of Modern Physics, vol. 51, no. 4, p. 863, 1979.

[22] M. Planck, “On the law of distribution of energy in the normal spectrum,” Annalen der physik, vol. 4, no. 553, p. 1, 1901.

[23] Z. Peng, S. De Graaf, J. Tsai, and O. Astafiev, “Tuneable on-demand single-photon source in the microwave range,” Nature communications, vol. 7, p. 12588, 2016.

[24] B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New approaches to nanofabrication: molding, printing, and other techniques,” Chemical reviews, vol. 105, no. 4, pp. 1171–1196, 2005.

[25] J. S. Bell, “The debate on the significance of his contributions to the foundations of quantum mechanics, Bells Theorem and the Foundations of Modern Physics (A. van der Merwe, F. Selleri, and G. Tarozzi, eds.),” 1992.

[26] D. J. Griffiths and D. F. Schroeter, Introduction to quantum mechanics. Cambridge University Press, 2018.

[27] C. Pickover, Archimedes to Hawking: Laws of science and the great minds behind them. Oxford University Press, 2008.

[28] R. Resnick, J. Walker, and D. Halliday, Fundamentals of physics. John Wiley, 1988.

[29] S. Ramo, J. R. Whinnery, and T. Duzer van, Fields and waves in communication electronics, Third Edition. John Wiley & Sons, Inc., 1995, also 1965, 1984.

[30] J. L. De Lagrange, “Recherches d’arithmétique,” Nouveaux Mémoires de l’Académie de Berlin, 1773.

[31] J. A. Kong, Electromagnetic Wave Theory. EMW Publishing, 2008, also 1985.

[32] H. M. Schey, Div, grad, curl, and all that: an informal text on vector calculus. WW Norton New York, 2005.

[33] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman lectures on physics, Vols. I, II, & III: The new millennium edition. Basic books, 2011, also 1963, 2006, vol. 1,2,3.

[34] W. C. Chew, Waves and fields in inhomogeneous media. IEEE Press, 1995, also 1990.

[35] V. J. Katz, “The history of Stokes’ theorem,” Mathematics Magazine, vol. 52, no. 3, pp. 146–156, 1979.

[36] W. K. Panofsky and M. Phillips, Classical electricity and magnetism. Courier Corporation, 2005.

[37] T. Lancaster and S. J. Blundell, Quantum field theory for the gifted amateur. OUP Oxford, 2014.

[38] W. C. Chew, “Fields and waves: Lecture notes for ECE 350 at UIUC,” https://engineering.purdue.edu/wcchew/ece350.html, 1990.

[39] C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.

[40] J. M. Crowley, Fundamentals of applied electrostatics. Krieger Publishing Company, 1986.

[41] C. Balanis, Advanced Engineering Electromagnetics. Hoboken, NJ, USA: Wiley, 2012.

[42] J. D. Jackson, Classical electrodynamics. John Wiley & Sons, 1999.

[43] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volumes 1 and 2. Interscience Publ., 1962.

[44] L. Esaki and R. Tsu, “Superlattice and negative differential conductivity in semiconductors,” IBM Journal of Research and Development, vol. 14, no. 1, pp. 61–65, 1970.

[45] E. Kudeki and D. C. Munson, Analog Signals and Systems. Upper Saddle River, NJ, USA: Pearson Prentice Hall, 2009.

[46] A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing. Pearson Education, 2014.

[47] R. F. Harrington, Time-harmonic electromagnetic fields. McGraw-Hill, 1961.

[48] E. C. Jordan and K. G. Balmain, Electromagnetic waves and radiating systems. Prentice-Hall, 1968.

[49] G. Agarwal, D. Pattanayak, and E. Wolf, “Electromagnetic fields in spatially dispersive media,” Physical Review B, vol. 10, no. 4, p. 1447, 1974.

[50] S. L. Chuang, Physics of photonic devices. John Wiley & Sons, 2012, vol. 80.

[51] B. E. Saleh and M. C. Teich, Fundamentals of photonics. John Wiley & Sons, 2019.

[52] M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Elsevier, 2013, also 1959 to 1986.

[53] R. W. Boyd, Nonlinear optics. Elsevier, 2003.

[54] Y.-R. Shen, The principles of nonlinear optics. New York, Wiley-Interscience, 1984.

[55] N. Bloembergen, Nonlinear optics. World Scientific, 1996.

[56] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of electric machinery. McGraw-Hill New York, 1986.

[57] A. E. Fitzgerald, C. Kingsley, S. D. Umans, and B. James, Electric machinery. McGraw-Hill New York, 2003, vol. 5.

[58] M. A. Brown and R. C. Semelka, MRI.: Basic Principles and Applications. John Wiley & Sons, 2011.

[59] C. A. Balanis, Advanced engineering electromagnetics. John Wiley & Sons, 1999, also 1989.

[60] Wikipedia, “Lorentz force,” https://en.wikipedia.org/wiki/Lorentz_force/, accessed: 2019-09-06.

[61] R. O. Dendy, Plasma physics: an introductory course. Cambridge University Press, 1995.

[62] P. Sen and W. C. Chew, “The frequency dependent dielectric and conductivity response of sedimentary rocks,” Journal of microwave power, vol. 18, no. 1, pp. 95–105, 1983.

[63] D. A. Miller, Quantum Mechanics for Scientists and Engineers. Cambridge, UK: Cambridge University Press, 2008.

[64] W. C. Chew, “Quantum mechanics made simple: Lecture notes for ECE 487 at UIUC,” http://wcchew.ece.illinois.edu/chew/course/QMAll20161206.pdf, 2016.

[65] B. G. Streetman and S. Banerjee, Solid state electronic devices. Prentice hall Englewood Cliffs, NJ, 1995.

[66] Smithsonian, “This 1600-year-old goblet shows that the romans were nanotechnology pioneers,” https://www.smithsonianmag.com/history/this-1600-year-old-goblet-shows-that-the-romans-were-nanotechnology-pioneers-787224/, accessed: 2019-09-06.

[67] K. G. Budden, Radio waves in the ionosphere. Cambridge University Press, 2009.

[68] R. Fitzpatrick, Plasma physics: an introduction. CRC Press, 2014.

[69] G. Strang, Introduction to linear algebra. Wellesley-Cambridge Press Wellesley, MA, 1993, vol. 3.

[70] K. C. Yeh and C.-H. Liu, “Radio wave scintillations in the ionosphere,” Proceedings of the IEEE, vol. 70, no. 4, pp. 324–360, 1982.

[71] J. Kraus, Electromagnetics. McGraw-Hill, 1984, also 1953, 1973, 1981.

[72] Wikipedia, “Circular polarization,” https://en.wikipedia.org/wiki/Circular_polarization.

[73] Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Advances in Optics and Photonics, vol. 1, no. 1, pp. 1–57, 2009.

[74] H. Haus, Electromagnetic Noise and Quantum Optical Measurements, ser. Advanced Texts in Physics. Springer Berlin Heidelberg, 2000.

[75] W. C. Chew, “Lectures on theory of microwave and optical waveguides, for ECE 531 at UIUC,” https://engineering.purdue.edu/wcchew/course/tgwAll20160215.pdf, 2016.

[76] L. Brillouin, Wave propagation and group velocity. Academic Press, 1960.

[77] R. Plonsey and R. E. Collin, Principles and applications of electromagnetic fields. McGraw-Hill, 1961.

[78] M. N. Sadiku, Elements of electromagnetics. Oxford University Press, 2014.

[79] A. Wadhwa, A. L. Dal, and N. Malhotra, “Transmission media,” https://www.slideshare.net/abhishekwadhwa786/transmission-media-9416228.

[80] P. H. Smith, “Transmission line calculator,” Electronics, vol. 12, no. 1, pp. 29–31, 1939.

[81] F. B. Hildebrand, Advanced calculus for applications. Prentice-Hall, 1962.

[82] J. Schutt-Aine, “Experiment02-coaxial transmission line measurement using slotted line,” http://emlab.uiuc.edu/ece451/ECE451Lab02.pdf.

[83] D. M. Pozar, E. J. K. Knapp, and J. B. Mead, “ECE 584 microwave engineering laboratory notebook,” http://www.ecs.umass.edu/ece/ece584/ECE584 lab manual.pdf, 2004.

[84] R. E. Collin, Field theory of guided waves. McGraw-Hill, 1960.

[85] Q. S. Liu, S. Sun, and W. C. Chew, “A potential-based integral equation method for low-frequency electromagnetic problems,” IEEE Transactions on Antennas and Propagation, vol. 66, no. 3, pp. 1413–1426, 2018.

[86] Wikipedia, “Snell’s law,” https://en.wikipedia.org/wiki/Snell’s_law.

[87] G. Tyras, Radiation and propagation of electromagnetic waves. Academic Press, 1969.

[88] L. Brekhovskikh, Waves in layered media. Academic Press, 1980.

[89] Scholarpedia, “Goos-hanchen effect,” http://www.scholarpedia.org/article/Goos-Hanchen_effect.

[90] K. Kao and G. A. Hockham, “Dielectric-fibre surface waveguides for optical frequencies,” in Proceedings of the Institution of Electrical Engineers, vol. 113, no. 7. IET, 1966, pp. 1151–1158.

[91] E. Glytsis, “Slab waveguide fundamentals,” http://users.ntua.gr/eglytsis/IO/SlabWaveguides_p.pdf, 2018.

[92] Wikipedia, “Optical fiber,” https://en.wikipedia.org/wiki/Optical_fiber.

[93] Atlantic Cable, “1869 indo-european cable,” https://atlantic-cable.com/Cables/1869IndoEur/index.htm.

[94] Wikipedia, “Submarine communications cable,” https://en.wikipedia.org/wiki/Submarine_communications_cable.

[95] D. Brewster, “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Philosophical Transactions of the Royal Society of London, vol. 105, pp. 125–159, 1815.

[96] Wikipedia, “Brewster’s angle,” https://en.wikipedia.org/wiki/Brewster’s_angle.

[97] H. Raether, “Surface plasmons on smooth surfaces,” in Surface plasmons on smooth and rough surfaces and on gratings. Springer, 1988, pp. 4–39.

[98] E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift für Naturforschung A, vol. 23, no. 12, pp. 2135–2136, 1968.

[99] Wikipedia, “Surface plasmon,” https://en.wikipedia.org/wiki/Surface_plasmon.

[100] Wikimedia, “Gaussian wave packet,” https://commons.wikimedia.org/wiki/File:Gaussian_wave_packet.svg.

[101] Wikipedia, “Charles K. Kao,” https://en.wikipedia.org/wiki/Charles_K._Kao.

[102] H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Physical Review, vol. 83, no. 1, p. 34, 1951.

[103] R. Kubo, “The fluctuation-dissipation theorem,” Reports on progress in physics, vol. 29, no. 1, p. 255, 1966.

[104] C. Lee, S. Lee, and S. Chuang, “Plot of modal field distribution in rectangular and circular waveguides,” IEEE transactions on microwave theory and techniques, vol. 33, no. 3, pp. 271–274, 1985.

[105] W. C. Chew, Waves and Fields in Inhomogeneous Media. IEEE Press, 1996.

[106] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Corporation, 1965, vol. 55.

[107] ——, “Handbook of mathematical functions: with formulas, graphs, and mathematical tables,” http://people.math.sfu.ca/∼cbm/aands/index.htm.

[108] W. C. Chew, W. Sha, and Q. I. Dai, “Green’s dyadic, spectral function, local density of states, and fluctuation dissipation theorem,” arXiv preprint arXiv:1505.01586, 2015.

[109] Wikipedia, “Very Large Array,” https://en.wikipedia.org/wiki/Very_Large_Array.

[110] C. A. Balanis and E. Holzman, “Circular waveguides,” Encyclopedia of RF and Microwave Engineering, 2005.

[111] M. Al-Hakkak and Y. Lo, “Circular waveguides with anisotropic walls,” Electronics Letters, vol. 6, no. 24, pp. 786–789, 1970.

[112] Wikipedia, “Horn Antenna,” https://en.wikipedia.org/wiki/Horn_antenna.

[113] P. Silvester and P. Benedek, “Microstrip discontinuity capacitances for right-angle bends, t junctions, and crossings,” IEEE Transactions on Microwave Theory and Techniques, vol. 21, no. 5, pp. 341–346, 1973.

[114] R. Garg and I. Bahl, “Microstrip discontinuities,” International Journal of Electronics Theoretical and Experimental, vol. 45, no. 1, pp. 81–87, 1978.

[115] P. Smith and E. Turner, “A bistable fabry-perot resonator,” Applied Physics Letters, vol. 30, no. 6, pp. 280–281, 1977.

[116] A. Yariv, Optical electronics. Saunders College Publ., 1991.

[117] Wikipedia, “Klystron,” https://en.wikipedia.org/wiki/Klystron.

[118] ——, “Magnetron,” https://en.wikipedia.org/wiki/Cavity_magnetron.

[119] ——, “Absorption Wavemeter,” https://en.wikipedia.org/wiki/Absorption_wavemeter.

[120] W. C. Chew, M. S. Tong, and B. Hu, “Integral equation methods for electromagnetic and elastic waves,” Synthesis Lectures on Computational Electromagnetics, vol. 3, no. 1, pp. 1–241, 2008.

[121] A. D. Yaghjian, “Reflections on Maxwell’s treatise,” Progress In Electromagnetics Research, vol. 149, pp. 217–249, 2014.

[122] L. Nagel and D. Pederson, “Simulation program with integrated circuit emphasis,” in Midwest Symposium on Circuit Theory, 1973.

420 Electromagnetic Field Theory

[123] S. A. Schelkunoff and H. T. Friis, Antennas: theory and practice. Wiley New York, 1952, vol. 639.

[124] H. G. Schantz, “A brief history of uwb antennas,” IEEE Aerospace and Electronic Systems Magazine, vol. 19, no. 4, pp. 22–26, 2004.

[125] E. Kudeki, “Fields and Waves,” http://remote2.ece.illinois.edu/∼erhan/FieldsWaves/ECE350lectures.html.

[126] Wikipedia, “Antenna Aperture,” https://en.wikipedia.org/wiki/Antenna_aperture.

[127] C. A. Balanis, Antenna theory: analysis and design. John Wiley & Sons, 2016.

[128] R. W. P. King, G. S. Smith, M. Owens, and T. Wu, “Antennas in matter: Fundamentals, theory, and applications,” NASA STI/Recon Technical Report A, vol. 81, 1981.

[129] H. Yagi and S. Uda, “Projector of the sharpest beam of electric waves,” Proceedings of the Imperial Academy, vol. 2, no. 2, pp. 49–52, 1926.

[130] Wikipedia, “Yagi-Uda Antenna,” https://en.wikipedia.org/wiki/Yagi-Uda_antenna.

[131] Antenna-theory.com, “Slot Antenna,” http://www.antenna-theory.com/antennas/aperture/slot.php.

[132] A. D. Olver and P. J. Clarricoats, Microwave horns and feeds. IET, 1994, vol. 39.

[133] B. Thomas, “Design of corrugated conical horns,” IEEE Transactions on Antennas and Propagation, vol. 26, no. 2, pp. 367–372, 1978.

[134] P. J. B. Clarricoats and A. D. Olver, Corrugated horns for microwave antennas. IET, 1984, no. 18.

[135] P. Gibson, “The vivaldi aerial,” in 1979 9th European Microwave Conference. IEEE, 1979, pp. 101–105.

[136] Wikipedia, “Vivaldi Antenna,” https://en.wikipedia.org/wiki/Vivaldi_antenna.

[137] ——, “Cassegrain Antenna,” https://en.wikipedia.org/wiki/Cassegrain_antenna.

[138] ——, “Cassegrain Reflector,” https://en.wikipedia.org/wiki/Cassegrain_reflector.

[139] W. A. Imbriale, S. S. Gao, and L. Boccia, Space antenna handbook. John Wiley & Sons, 2012.

[140] J. A. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Transactions on Antennas and Propagation, vol. 49, no. 10, pp. 1403–1410, 2001.

[141] D.-C. Chang and M.-C. Huang, “Microstrip reflectarray antenna with offset feed,” Electronics Letters, vol. 28, no. 16, pp. 1489–1491, 1992.

421 Quantum Coherent State of Light

[142] G. Minatti, M. Faenzi, E. Martini, F. Caminita, P. De Vita, D. González-Ovejero, M. Sabbadini, and S. Maci, “Modulated metasurface antennas for space: Synthesis, analysis and realizations,” IEEE Transactions on Antennas and Propagation, vol. 63, no. 4, pp. 1288–1300, 2014.

[143] X. Gao, X. Han, W.-P. Cao, H. O. Li, H. F. Ma, and T. J. Cui, “Ultrawideband and high-efficiency linear polarization converter based on double v-shaped metasurface,” IEEE Transactions on Antennas and Propagation, vol. 63, no. 8, pp. 3522–3530, 2015.

[144] D. De Schweinitz and T. L. Frey Jr, “Artificial dielectric lens antenna,” Nov. 13 2001, US Patent 6,317,092.

[145] K.-L. Wong, “Planar antennas for wireless communications,” Microwave Journal, vol. 46, no. 10, pp. 144–145, 2003.

[146] H. Nakano, M. Yamazaki, and J. Yamauchi, “Electromagnetically coupled curl antenna,” Electronics Letters, vol. 33, no. 12, pp. 1003–1004, 1997.

[147] K. Lee, K. Luk, K.-F. Tong, S. Shum, T. Huynh, and R. Lee, “Experimental and simulation studies of the coaxially fed U-slot rectangular patch antenna,” IEE Proceedings-Microwaves, Antennas and Propagation, vol. 144, no. 5, pp. 354–358, 1997.

[148] K. Luk, C. Mak, Y. Chow, and K. Lee, “Broadband microstrip patch antenna,” Electronics letters, vol. 34, no. 15, pp. 1442–1443, 1998.

[149] M. Bolic, D. Simplot-Ryl, and I. Stojmenovic, RFID systems: research trends and challenges. John Wiley & Sons, 2010.

[150] D. M. Dobkin, S. M. Weigand, and N. Iyer, “Segmented magnetic antennas for near-field UHF RFID,” Microwave Journal, vol. 50, no. 6, p. 96, 2007.

[151] Z. N. Chen, X. Qing, and H. L. Chung, “A universal UHF RFID reader antenna,” IEEE transactions on microwave theory and techniques, vol. 57, no. 5, pp. 1275–1282, 2009.

[152] C.-T. Chen, Linear system theory and design. Oxford University Press, Inc., 1998.

[153] S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Historia mathematica, vol. 19, no. 4, pp. 385–401, 1992.

[154] A. Ishimaru, Electromagnetic wave propagation, radiation, and scattering from fundamentals to applications. Wiley Online Library, 2017, also 1991.

[155] A. E. H. Love, “I. the integration of the equations of propagation of electric waves,” Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, vol. 197, no. 287-299, pp. 1–45, 1901.

[156] Wikipedia, “Christiaan Huygens,” https://en.wikipedia.org/wiki/Christiaan_Huygens.

[157] ——, “George Green (mathematician),” https://en.wikipedia.org/wiki/George_Green(mathematician).

422 Electromagnetic Field Theory

[158] C.-T. Tai, Dyadic Greens Functions in Electromagnetic Theory. PA: International Textbook, Scranton, 1971.

[159] ——, Dyadic Green functions in electromagnetic theory. Institute of Electrical & Electronics Engineers (IEEE), 1994.

[160] W. Franz, “Zur formulierung des huygensschen prinzips,” Zeitschrift für Naturforschung A, vol. 3, no. 8-11, pp. 500–506, 1948.

[161] J. A. Stratton, Electromagnetic Theory. McGraw-Hill Book Company, Inc., 1941.

[162] J. D. Jackson, Classical Electrodynamics. John Wiley & Sons, 1962.

[163] W. Meissner and R. Ochsenfeld, “Ein neuer effekt bei eintritt der supraleitfähigkeit,” Naturwissenschaften, vol. 21, no. 44, pp. 787–788, 1933.

[164] Wikipedia, “Superconductivity,” https://en.wikipedia.org/wiki/Superconductivity.

[165] D. Sievenpiper, L. Zhang, R. F. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Transactions on Microwave Theory and techniques, vol. 47, no. 11, pp. 2059–2074, 1999.

[166] Wikipedia, “Snell’s law,” https://en.wikipedia.org/wiki/Snell’s_law.

[167] H. Lamb, “On sommerfeld’s diffraction problem; and on reflection by a parabolic mirror,” Proceedings of the London Mathematical Society, vol. 2, no. 1, pp. 190–203, 1907.

[168] W. J. Smith, Modern optical engineering. McGraw-Hill New York, 1966, vol. 3.

[169] D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive optics: design, fabrication, and test. Spie Press Bellingham, WA, 2004, vol. 62.

[170] J. B. Keller and H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” JOSA, vol. 40, no. 1, pp. 48–52, 1950.

[171] G. A. Deschamps, “Ray techniques in electromagnetics,” Proceedings of the IEEE, vol. 60, no. 9, pp. 1022–1035, 1972.

[172] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proceedings of the IEEE, vol. 62, no. 11, pp. 1448–1461, 1974.

[173] R. Kouyoumjian, “The geometrical theory of diffraction and its application,” in Numerical and Asymptotic Techniques in Electromagnetics. Springer, 1975, pp. 165–215.

[174] S.-W. Lee and G. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Transactions on Antennas and Propagation, vol. 24, no. 1, pp. 25–34, 1976.

[175] Wikipedia, “Fermat’s principle,” https://en.wikipedia.org/wiki/Fermat’s_principle.

423 Quantum Coherent State of Light

[176] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science, vol. 334, no. 6054, pp. 333–337, 2011.

[177] A. Sommerfeld, Partial differential equations in physics. Academic Press, 1949, vol. 1.

[178] R. Haberman, Elementary applied partial differential equations. Prentice Hall Englewood Cliffs, NJ, 1983, vol. 987.

[179] G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electronics letters, vol. 7, no. 23, pp. 684–685, 1971.

[180] J. Enderlein and F. Pampaloni, “Unified operator approach for deriving hermite–gaussian and laguerre–gaussian laser modes,” JOSA A, vol. 21, no. 8, pp. 1553–1558, 2004.

[181] D. L. Andrews, Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces. Academic Press, 2011.

[182] J. W. Strutt, “Xv. on the light from the sky, its polarization and colour,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 41, no. 271, pp. 107–120, 1871.

[183] L. Rayleigh, “X. on the electromagnetic theory of light,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 12, no. 73, pp. 81–101, 1881.

[184] R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Transactions on Antennas and Propagation, vol. 36, no. 8, pp. 1078–1087, 1988.

[185] S. Sun, Y. G. Liu, W. C. Chew, and Z. Ma, “Calderón multiplicative preconditioned efie with perturbation method,” IEEE Transactions on Antennas and Propagation, vol. 61, no. 1, pp. 247–255, 2012.

[186] G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Annalen der physik, vol. 330, no. 3, pp. 377–445, 1908.

[187] Wikipedia, “Mie scattering,” https://en.wikipedia.org/wiki/Mie_scattering.

[188] R. E. Collin, Foundations for microwave engineering. John Wiley & Sons, 2007, also 1966.

[189] L. B. Felsen and N. Marcuvitz, Radiation and scattering of waves. John Wiley & Sons, 1994, also 1973, vol. 31.

[190] P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der physik, vol. 369, no. 3, pp. 253–287, 1921.

[191] E. Whitaker and G. Watson, A Course of Modern Analysis. Cambridge Mathematical Library, 1927.

424 Electromagnetic Field Theory

[192] A. Sommerfeld, Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie ¨ . Verlag der K¨oniglich Bayerischen Akademie der Wissenschaften, 1909.

[193] J. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics, vol. 37, no. 6, pp. 985–996, 1972.

[194] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation, vol. 14, no. 3, pp. 302–307, 1966.

[195] A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion, vol. 10, no. 6, pp. 547–582, 1988.

[196] A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method. Artech house, 2005, also 1995.

[197] W. Yu, R. Mittra, T. Su, Y. Liu, and X. Yang, Parallel finite-difference time-domain method. Artech House Norwood, 2006.

[198] D. Potter, “Computational physics,” 1973.

[199] W. F. Ames, Numerical methods for partial differential equations. Academic press, 2014, also 1977.

[200] K. W. Morton, Revival: Numerical Solution Of Convection-Diffusion Problems (1996). CRC Press, 2019.

[201] K. Aki and P. G. Richards, Quantitative seismology, 2002.

[202] W. C. Chew, “Electromagnetic theory on a lattice,” Journal of Applied Physics, vol. 75, no. 10, pp. 4843–4850, 1994.

[203] J. v. Neumann, Mathematische Grundlagen der Quantenmechanik, Berlın. Springer, New York, Dover Publications, 1943.

[204] R. Courant, K. Friedrichs, and H. Lewy, “Uber die partiellen differenzengleichungen der mathematischen physik,” Mathematische annalen, vol. 100, no. 1, pp. 32–74, 1928.

[205] J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of computational physics, vol. 114, no. 2, pp. 185–200, 1994.

[206] W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microwave and optical technology letters, vol. 7, no. 13, pp. 599–604, 1994.

[207] W. C. Chew, J. Jin, and E. Michielssen, “Complex coordinate system as a generalized absorbing boundary condition,” in IEEE Antennas and Propagation Society International Symposium 1997. Digest, vol. 3. IEEE, 1997, pp. 2060–2063.

425 Quantum Coherent State of Light

[208] W. C. H. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, 2000.

[209] G. C. Hsiao and W. L. Wendland, Boundary integral equations. Springer, 2008.

[210] K. F. Warnick, Numerical analysis for electromagnetic integral equations. Artech House, 2008.

[211] M. M. Botha, “Solving the volume integral equations of electromagnetic scattering,” Journal of Computational Physics, vol. 218, no. 1, pp. 141–158, 2006.

[212] P. K. Banerjee and R. Butterfield, Boundary element methods in engineering science. McGraw-Hill London, 1981, vol. 17.

[213] O. C. Zienkiewicz, R. L. Taylor, P. Nithiarasu, and J. Zhu, The finite element method. McGraw-Hill London, 1977, vol. 3.

[214] J.-F. Lee, R. Lee, and A. Cangellaris, “Time-domain finite-element methods,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 3, pp. 430–442, 1997.

[215] J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite element method electromagnetics: antennas, microwave circuits, and scattering applications. John Wiley & Sons, 1998, vol. 6.

[216] J.-M. Jin, The finite element method in electromagnetics. John Wiley & Sons, 2015.

[217] G. Strang, Linear algebra and its applications. Academic Press, 1976.

[218] Cramer and Gabriel, Introduction a l’analyse des lignes courbes algebriques par Gabriel Cramer... chez les freres Cramer & Cl. Philibert, 1750.

[219] J. A. Schouten, Tensor analysis for physicists. Courier Corporation, 1989.

[220] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press, 2007.

[221] Wikipedia, “John Dalton,” https://en.wikipedia.org/wiki/John_Dalton.

[222] ——, “Max Planck,” https://en.wikipedia.org/wiki/Max_Planck.

[223] ——, “Photoelectric effect,” https://en.wikipedia.org/wiki/Photoelectric_effect.

[224] B. Bapat, “Newton’s rings,” http://www.iiserpune.ac.in/∼bhasbapat/phy221_files/NewtonsRing.pdf.

[225] Wikipedia, “Double-slit experiment,” https://en.wikipedia.org/wiki/Double-slit_experiment.

[226] Shmoop.Com, “Young’s double-slit,” https://www.shmoop.com/optics/young-double-slit.html.

[227] Wikipedia, “Louis de Broglie,” https://en.wikipedia.org/wiki/Louis_de_Broglie.

426 Electromagnetic Field Theory

[228] ——, “Newton’s laws of motion,” https://en.wikipedia.org/wiki/Newton’s_laws_of_motion.

[229] ——, “Quantum electrodynamics,” https://en.wikipedia.org/wiki/Quantum_electrodynamics.

[230] C. Christopoulos, The transmission-line modeling method: TLM. IEEE New York, 1995, vol. 221.

[231] A. E. Ruehli, G. Antonini, and L. Jiang, Circuit oriented electromagnetic modeling using the PEEC techniques. Wiley Online Library, 2017.

[232] Wikipedia, “William Rowan Hamilton,” https://en.wikipedia.org/wiki/William_Rowan_Hamilton.

[233] W. C. Chew, A. Y. Liu, C. Salazar-Lazaro, D. Na, and W. E. I. Sha, “Hamilton equation, commutator, and energy conservation,” Quantum Report, in press, 2019.

[234] Wikipedia, “Gaussian beam,” https://en.wikipedia.org/wiki/Gaussian_beam.

[235] M. Kira and S. W. Koch, Semiconductor quantum optics. Cambridge University Press, 2011.

[236] Wikipedia, “Quantum harmonic oscillator,” https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator.

[237] ——, “E.C. George Sudarshan,” https://en.wikipedia.org/wiki/E._C._George_Sudarshan.

[238] ——, “Roy J. Glauber,” https://en.wikipedia.org/wiki/Roy_J._Glauber.

[239] C. Gerry, P. Knight, and P. L. Knight, Introductory quantum optics. Cambridge University Press, 2005.