Open access peer-reviewed chapter

# Green Function

By Jing Huang

Submitted: October 18th 2016Reviewed: February 21st 2017Published: June 14th 2017

DOI: 10.5772/68028

## Abstract

Both the scalar Green function and the dyadic Green function of an electromagnetic field and the transform from the scalar to dyadic Green function are introduced. The Green function of a transmission line and the propagators are also presented in this chapter.

### Keywords

• Green function
• boundary condition
• scatter
• propagator
• convergence

## 1. Introduction

In 1828, Green introduced a function, which he called a potential, for calculating the distribution of a charge on a surface bounding a region in Rn in the presence of external electromagnetic forces. The Green function has been an interesting topic in modern physics and engineering, especially for the electromagnetic theory in various source distributions (charge, current, and magnetic current), various construct conductors, and dielectric. Even though most problems can be solved without the use of Green functions, the symbolic simplicity with which they could be used to express relationships makes the formulations of many problems simpler and more compact. Moreover, it is easier to conceptualize many problems; especially the dyadic Green function is generalized to layered media of planar, cylindrical, and spherical configurations.

## 2. Definition of Green function

### 2.1. Mathematics definition

For the linear operator, there are: L^x=f(t), t > 0;

x(t)|t=0=y0;x(n)(t)|t=0=ynE1

Rewriting Eq. (1) as:

L^x=f(t)δ(tt)dtE2

Defining the Green function as:

L^G(t,t)=δ(tt)E3

So, the solution of Eq. (1) is:

x(t)=f(t)G(t,t)dtE4

We give several types of Green functions [1]

 L^=−(d2dt2+2γddt+ω02) G(t,t′)=12π∫−∞+∞exp[−i(t−t′)k]k2+2iγk−ω02dk L^=−[f0(t)d2dt2+f1(t)ddt+f2(t)) G(t,t′)=-Ψ1(t)Ψ2(t′)−Ψ2(t)Ψ1(t′)f0(t′)[Ψ1(t′)Ψ˙2(t′)−Ψ˙1(t′)Ψ2(t′)] L^=−ddt[(1−t2)ddt] G(t,t′)=12+∑n=1∞1n(n+1)⋅2n+12Pn(t)Pn(t′)

## 3. The scalar Green function

### 3.1. The scalar Green function of an electromagnetic field

The Green function of a wave equation is the solution of the wave equation for a point source [2]. And when the solution to the wave equation due to a point source is known, the solution due to a general source can be obtained by the principle of linear superposition (see Figure 1).

This is merely a result of the linearity of the wave equation, and that a general source is just a linear superposition of point sources. For example, to obtain the solution to the scalar wave equation in V in Figure 1

(2+k2)ϕ(r)=s(r)E5

we first seek the Green function in the same V, which is the solution to the following equation:

(2+k2)g(r,r')=δ(r-r')E6

Given g (r, r′), φ(r) can be found easily from the principle of linear superposition, since g (r, r′) is the solution to Eq. (5) with a point source on the right-hand side. To see this more clearly, note that an arbitrary source s(r) is just

s(r)=dr's(r')δ(rr')E7

which is actually a linear superposition of point sources in mathematical terms. Consequently, the solution to Eq. (5) is just

ϕ(r)=Vdr'g(r,r')s(r')E8

which is an integral linear superposition of the solution of Eq. (6). Moreover, it can be seen that g(r, r′) ≡ g(r′, r,) from reciprocity irrespective of the shape of V.

To find the solution of Eq. (6) for an unbounded, homogeneous medium, one solves it in spherical coordinates with the origin at r'. By so doing, Eq. (6) becomes

(2+k2)g(r)=δ(x)δ(y)δ(z)E9

But due to the spherical symmetry of a point source, g(r) must also be spherically symmetric. Then, for r ≠ 0, adopt the proper coordinate origin (the vector r is replaced by the scalar r), the homogeneous, spherically symmetric solution to Eq. (9) is given by

g(r)=c1eikrr+c2eikrrE10

Since sources are absent at infinity, physical grounds then imply that only an outgoing solution can exist; hence,

g(r)=ceikrrE11

The constant c is found by matching the singularities at the origin on both sides of Eq. (9). To do this, we substitute Eq. (11) into Eq. (9) and integrate Eq. (9) over a small volume about the origin to yield

ΔVdVceikrr+ΔVdVk2ceikrr=1E12

Note that the second integral vanishes when ∆V → 0 because dV = 4πr2dr. Moreover, the first integral in Eq. (12) can be converted into a surface integral using Gauss theorem to obtain

limr04πr2ddrceikrr=1E13

or c = 1/(4π).

The solution to Eq. (6) must depend only on rr′. Therefore, in general,

g(r,r')=g(rr')=eik(rr')4π(rr')E14

implying that g(r, r') is translationally invariant for unbounded, homogeneous media. Consequently, the solution to Eq. (5), from Eq. (9), is then

ϕ(r)=Vdr'eik(rr')4π(rr')s(r')E15

Once ϕ(r) and n^ϕ(r)are known on S, then ϕ(r′) away from S could be found

ϕ(r)=SdSn^[g(r,r)ϕ(r)ϕ(r)g(r,r)]E16

### 3.2. The scalar Green functions of one-dimensional transmission lines

We consider a transmission line excited by a distributed current source, K(x), as sketched in Figure 2. The line may be finite or infinite, and it may be terminated at either end with impedance or by another line [3]. For a harmonically oscillating current source K(x), the voltage and the current on the line satisfy the following pair of equations:

dV(x)dx=iωLI(x)E17
dI(x)dx=iωCV(x)+K(x)E18

L and C denote, respectively, the distributed inductance and capacitance of the line.

By eliminating I(x) between Eq. (17) and Eq. (18), there is

d2V(x)dx2+k2V(x)=iωLK(x)E19

where k=ωLCdenotes the propagation constant of the line. Eq. (19) has been designated as an inhomogeneous one-dimensional scalar wave equation.

The Green function pertaining to a one-dimensional scalar wave equation of the form of Eq. (19), denoted by g(x, x′), is a solution of the Eq. (9). The solution for g(x, x′) is not completely determined unless there are two boundary conditions which the function must satisfy at the extremities of the spatial domain in which the function is defined. The boundary conditions which must be satisfied by g(x, x′) are the same as those dictated by the original function which we intend to determine, namely, V(x) in the present case. For this reason, the Green functions are classified according to the boundary conditions, which they must obey. Some of the typical ones (for the transmission line) are illustrated in Figure 3.

In general, the subscript 0 designates infinite domain so that we have outgoing waves at x±, often called the radiation condition. Subscript 1 means that one of the boundary conditions satisfies the so-called Dirichlet condition, while the other satisfies the radiation condition. When one of the boundary conditions satisfies the so-called Neumann condition, we use subscript 2. Subscript 3 is reserved for the mixed type. Actually, we should have used a double subscript for two distinct boundary conditions. For example, case (b) of Figure 3 should be denoted by g01, indicating that one radiation condition and one Dirichlet condition are involved. With such an understanding, the simplified notation should be acceptable.

In case (d), a superscript becomes necessary because we have two sets of line voltage and current (V1, I1) and (V2, I2) in this problem, and the Green function also has different forms in the two regions. The first superscript denotes the region where this function is defined, and the second superscript denotes the region where the source is located.

Let the domain of x corresponds to (x1, x2). The function g(x, x′) in Eq. (9) can represent any of the three types, g0, g1, and g2, illustrated in Figures 3a–c, respectively. The treatment of case (d) is slightly different, and it will be formulated later.

(a) By multiplying Eq. (19) by g(x, x′) and Eq. (9) by V(x) and taking the difference of the two resultant equations, we obtain

x1x2[V(x)d2g0(x,x)dx2g0(x,x)d2V(x,x)dx2]dx=x1x2V(x)δ(xx)dxiωLx1x2K(x)g0(x,x)dxE20

The first term at the right-hand side of the above equation is simply V(xl), and the term at the left-hand side can be simplified by integration by parts, which gives

V(x)=iωLx1x2g0(x,x)K(x)dxE21

If we use the unprimed variable x to denote the position of a field point, as usually is the case, Eq. (21) can be changed to [4]

V(x)=iωLx1x2g(x,x)K(x)dx=iωLx1x2g0(x,x)K(x)dxE22

The last identity is due to the symmetrical property of the Green function. The shifting of the primed and unprimed variables is often practiced in our work. For this reason, it is important to point out that g(x′, x), by definition, satisfies the Eq. (9).

The general solutions for Eq. (9) in the two regions (see Figure 3a) are

g0(x,x)={i/(2k)eik(xx),xxi/(2k)eik(xx),xxE23

The choice of the above functions is done with the proper satisfaction of boundary conditions at infinity. At x = x', the function must be continuous, and its derivative is discontinuous.

They are: [g0(x,x)]x0x+0=0, and [dg0(x,x)dx]x0x+0=-1

The physical interpretation of these two conditions is that the voltage at x' is continuous, but the difference of the line currents at x' must be equal to the source current.

(b) The choice of this type of function is done with the proper satisfaction of boundary conditions. At x = x', the function must be continuous, its derivative is discontinuous, and a Dirichlet condition is satisfied at x = 0.

g1(x,x)={ι/(2κ)[eικ(xx)eικ(x+x)],xxι/(2κ)[eικ(xx)eικ(x+x)],0xxE24

In view of Eq. (24), it can be interpreted as consisting of an incident and a scattered wave; that is

g1(x,x)=g0(x,x)+g1s(x,x)E25

where g1s(x,x)=i2keik(x+x).

Such a notion is not only physically useful, but mathematically it offers a shortcut to finding a composite Green function. It is called as the shortcut method or the method of scattering superposition.

(c) Similarly, the method of scattering superposition suggests that we can start with

g2(x,x)=g0(x,x)+AeikxE26

To satisfy the Neumann condition at x = 0, we require

[dg0(x,x)dx+ikAeikx]x=0=0E27

Hence

A=i2keikxE28
g2(x,x)=i/(2k){eik(xx)+eik(x+x),xxeik(xx)+eik(x+x),0xxE29

(d) In this case, we have two differential equations to start with

d2V1(x)dx2+k12V1(x)=iωL1K1(x),x0E30
d2V2(x)dx2+k22V2(x)=0,x0E31

It is assumed that the current source is located in region 1 (see Figure 3d). We introduce two Green functions of the third kind, denoted by g(11) (x, x') and g(21) (x, x'). g(21), the first number of the superscript corresponds to the region where the function is defined. The second number corresponds to the region where the source is located; then

d2g(11)(x,x)dx2+k12g(11)(x,x)=δ(xx),x0E32
d2g(21)(x,x)dx2+k22g(21)(x,x)=0,x0E33

At the junction corresponding to x = 0, g(11) and g(21) satisfy the boundary condition that

g(11)(x,x)x=0=g(21)(x,x)x=0E34
1L1dg(11)(x,x)dxx=0=1L2dg(21)(x,x)dxx=0E35

The last condition corresponds to the physical requirement that the current at the junction must be continuous. Again, by means of the method of scattering superposition, there are

g(11)(x,x)=g0(x,x)+gs(11)(x,x)=i2k1{eik1(xx)+Reik1(xx),xxeik1(xx)+Reik1(x+x),0xxE36
g(21)(x,x)=i2k1Tei(k2xk1x),x0E37

The characteristic impedance of the lines, respectively, is

z1=(L1C1)1/2,z2=(L2C2)1/2E38

By the boundary condition, there are

R=z2z1z2+z1,T=2z2z2+z1E39

Example: Green function solution of nonlinear Schrodinger equation in the time domain [5].

The nonlinear Schrodinger equation including nonresonant and resonant nonlinear items is:

Az+i2β22At216β33At3=a2A+i3k08nAeffχNR(3)|A|2A+ik0g(ω0)[1if(ω0)]2nAeffAtχR(3)(tτ)|A(τ)|2dτE40

Where A is the field, β2 and β3 are the second and third order dispersion, respectively. A(z) is the fiber absorption profile. k0=ω0/c, ω0 is the center frequency. Aeff is the effective core area. n is the refractive index.

f(ω1+ω2+ω3)=2(ω1+ω2+ω3)(1|Γ|)2(ω1+ω2+ω3)22|Γ|+|Γ|2E41
g(ω1+ω2+ω3)=[2(ω1+ω2+ω3)22|Γ|+|Γ|2]E42

where g(ω1 + ω2 + ω3) is the Raman gain and f(ω1 + ω2 + ω3) is the Raman nongain coefficient. Г is the attenuation coefficient.

The original nonlinear part is divided into the nonresonant and resonant susceptibility items χNR(3)and χR(3). The solution has the form:

A(z,t)=ϕ(t)eiEzE43

Then, there is:

12β22φt2+i6β33φt33k08nAeffχNR(3)|φ|2φk0g(ωs)[1if(ωs)]2nAeffφ+χN(3)(tτ)|φ(τ)|dτ=EφE44

Let:

H^0(t)=12β22t2+i6β33t3E45
V^(t)=3k08nAeffχNR(3)|φ|k0g(ωs)[1if(ωs)]2nAeff+χR(3)(tτ)|φ(τ)|2dτE46

and taking the operator V^(t)as a perturbation item, the eigenequation n=2kinn!βnnφTn=Eφis

12β22φT2+i6β33φT3=EφE47

Assuming E = 1, we get the corresponding characteristic equation:

12β2r2+β36r3=EE48

Its characteristic roots are r1,r2,r3. The solution can be represented as:

φ=c1φ1+c2φ2+c3φ3E49

where ϕm=exp(irmt),m=1,2,3, and c1,c2,c3 are determined by the initial pulse. The Green function of Eq. (47) is:

(EH^0(t))G0(t,t)=δ(tt)E50

Constructing the Green function as:

G0(t,t)={a1φ1+a2φ2+a3φ3,t>tb1φ1+b2φ2+b3φ3,t<tE51

At the point t = t′, there are:

a1φ1(t)+a2φ2(t)+a3φ3(t)=b1φ1(t)+b2φ2(t)+b3φ3(t)E52
a1φ1(t)+a2φ2(t)+a3φ3(t)=b1φ1(t)+b2φ2(t)+b3φ3(t)E53
a1φ1(t)+a2φ2(t)+a3φ3(t)b1φ1(t)b2φ2(t)b3φ3(t)=6i/β3E54

It is reasonable to let b1 = b2 = b3 = 0, then:

a1=φ2φ˙3φ˙2φ3W(t),a2=φ3φ˙1φ˙3φ1W(t),a3=φ1φ˙2φ˙1φ2W(t)E55
W(t)=|φ1φ2φ3φ1(1)φ2(1)φ3(1)φ1(2)φ2(2)φ3(2)|E56

Finally, the solution of Eq. (44) can be written with the eigenfunction and Green function:

φ(t)=φ(t)+G0(t,t)V(t)φ(t)dt=ϕ(t)+G0(t,t,E)V(t)ϕ(t)dt+dtG0(t,t,E)V(t)G0(t,t,E)V(t)φ(t)dt=ϕ(t)+G0(t,t,E)V(t)ϕ(t)dt+dtG0(t,t,E)V(t)G0(t,t,E)V(t)ϕ(t)dt++dtG0(t,t)V(t)G0(t,t)V(t)dttimes lG0(tl,tl+1)V(tl+1)φ(tl+1)dtl+1E57

The accuracy can be estimated by the last term of Eq. (57).

## 4. The dyadic Green function

### 4.1. The dyadic Green function for the electromagnetic field in a homogeneous isotropic medium

The Green function for the scalar wave equation could be used to find the dyadic Green function for the vector wave equation in a homogeneous, isotropic medium [3]. First, notice that the vector wave equation in a homogeneous, isotropic medium is

××E(r)k2E(r)=iωμJ(r)E58

Then, by using the fact that ××E(r)=-2E+Eand that E=ρ/ε=J/iωε, which follows from the continuity equation, we can rewrite Eq. (58) as

2E(r)k2E(r)=iωμ[I^+k2]J(r)E59

where I^is an identity operator. In Cartesian coordinates, there are actually three scalar wave equations embedded in the above vector equation, each of which can be solved easily in the manner of Eq. (4). Consequently,

E(r)=iωμVdr'g(r'r)[I^+''k2]J(r)E60

where g(r′r)is the unbounded medium scalar Green function. Moreover, by using the vector identities gf=fg+gfand gF=gF+(g)F, it can be shown that

Vdr'g(r'r)'f(r')=Vdr''g(r'r)f(r')E61

and

Vdr'['g(r'r)]'J(r')=Vdr'J(r')''g(r'r)E62

Hence, Eq. (60) can be rewritten as

E(r)=iωμVdr'J(r')[I^+''k2]g(r'r)E63

It can also be derived using scalar and vector potentials.

Alternatively, Eq. (63) can be written as

E(r)=iωμVdr'J(r')G^e(r',r)E64

where

G^e(r)=[I¯+''k2]g(r'r)E65

is a dyad known as the dyadic Green function for the electric field in an unbounded, homogeneous medium. (A dyad is a 3 × 3 matrix that transforms a vector to a vector. It is also a second rank tensor). Even though Eq. (64) is established for an unbounded, homogeneous medium, such a general relationship also exists in a bounded, homogeneous medium. It could easily be shown from reciprocity that

J1(r),G^e(r,r'),J2(r')=J2(r),G^e(r,r'),J1(r')=J1(r),G^et(r,r'),J2(r')E66

where

Ji(r),G^e(r,r'),Jj(r')=VVdr'drJi(r')G^e(r',r)Jj(r)E66a

is the relation between Ji and the electric field produced by Jj. Notice that the above equation implies [6]

G^et(r',r)=G^e(r,r')E66b

Then, by taking transpose of Eq. (66b), Eq. (64) becomes

E(r)=iωμVdr'G^e(r,r')J(r')E67

Alternatively, the dyadic Green function for an unbounded, homogeneous medium can also be written as

G^e(r,r')=1k2[××I^g(rr')I^δ(rr')]E68

By substituting Eq. (67) back into Eq. (58) and writing

J(r)=dr'I^δ(rr')J(r')E69

we can show quite easily that

××G^e(r,r')-k2G^e(r,r')=I^δ(rr')E70

Equation (64) or (67), due to the ∇∇ operator inside the integration operating on g(r′r), has a singularity of 1/|r′r|3 when r′r. Consequently, it has to be redefined in this case for it does not converge uniformly, specifically, when r is also in the source region occupied by J(r). Hence, at this point, the evaluation of Eq. (67) in a source region is undefined.

And as the vector analog of Eq. (16)

E(r)=SdS[n×E(r)×G^e(r,r)+iωμn×H(r)G^e(r,r)]E71

### 4.2. The boundary condition

The dyadic Green function is introduced mainly to formulate various canonical electromagnetic problems in a systematic manner to avoid treatments of many special cases which can be treated as one general problem [3, 7, 8]. Some typical problems are illustrated in Figure 4 where (a) shows a current source in the presence of a conducting sphere located in air, (b) shows a conducting cylinder with an aperture which is excited by some source inside the cylinder, (c) shows a rectangular waveguide with a current source placed inside the guide, and (d) shows two semi-infinite isotropic media in contact, such as air and “flat” earth with a current source placed in one of the regions.

Unless specified otherwise, we assume that for problems involving only one medium such as (a), (b), and (c) the medium is air, then the wave number k is equal to ω(μ0ε0)1/2=2π/λ. The electromagnetic fields in these cases are solutions of the wave Eq. (62) and

××H(r)k2H(r)=×J(r)E72

The fields must satisfy the boundary conditions required by these problems.

In general, using the notations G^eand G^mto denote, respectively, the electric and the magnetic dyadic Green functions; they are solutions of the dyadic differential equations

××G^e(r,r')-k2G^e(r,r')=I^δ(rr')E73
××G^m(r,r')-k2G^m(r,r')=×[I^δ(rr')]E74

is the same as Eq. (70), and there is

G^m=×G^eE75

(a) and (b): Electric dyadic Green function (the first kind, using the subscript 1 denotes G^e1,G^m1, and the subscript “0” represents the free-space condition that the environment does not have any scattering object) is required to satisfy the dyadic Dirichlet condition on Sd, namely,

n×G^e1=0,n×G^m1=0E76

So, for (a)

E(r)=drJ(r)G^e(r,r)E77

and for (b)

(c) the electric dyadic Green function is required to satisfy the dyadic boundary condition on Sd, namely,

n××G^e2=0n××G^m2=0E79
H(r)=drJ(r)×G^e(r,r)E80

(d) For problems involving two isotropic media such as the configuration shown in Figure 4d, there are two sets of fields [9]. The wave numbers in these two regions are denoted by k1=ω(μ1ε1)1/2and k2=ω(μ2ε2)1/2. There are four functions for the dyadic Green function of the electric type and another four functions for the magnetic type, denoted, respectively, by G^e11G^e12G^e21and G^e22, and G^m11G^m12G^m21and G^m22. The superscript notation in G^e11means that both the field point and the source point are located in region 1. For G^e21, it means that the field point is located in region 1 and the source point is located in region 2. A current source is located in region 1 only, and the two sets of wave equations are

××E1(r)k2E1(r)=iωμ1J1(r)E81
××H1(r)k2H1(r)=×J1(r)E82

and

××E2(r)k2E2(r)=0E83
××H2(r)k2H2(r)=0E84

There are

××G^e11(r,r')-k12G^e11(r,r')=I^δ(rr')E85
××G^e21(r,r')-k22G^e21(r,r')=0E86

At the interface, the electromagnetic field and the corresponding dyadic Green function satisfy the following boundary conditions

n×[G^e11G^e21]=0E87
n×[×G^e11/μ1×G^e21/μ2]=0E88

The electric fields are

E1(r)=iωμ1drJ(r)G^e11(r,r)E89
E2(r)=iωμ2drJ(r)G^e21(r,r)E90

## 5. Vector wave functions, L, M, and N

The vector wave functions are the building blocks of the eigenfunction expansions of various kinds of dyadic Green functions. These functions were first introduced by Hansen [1012] in formulating certain electromagnetic problems.Three kinds of vector wave functions, denoted by L, M, and N, are solutions of the homogeneous vector Helmholtz equation. To derive the eigenfunction expansion of the magnetic dyadic Green functions that are solenoidal and satisfy with the vector wave equation, the L functions are not needed. If we try to find eigenfunction expansion of the electric dyadic Green functions then the L functions are also needed.

A vector wave function, by definition, is an eigenfunction or a characteristic function, which is a solution of the homogeneous vector wave equation ××Fκ2F=0.

There are two independent sets of vector wave functions, which can be constructed using the characteristic function pertaining to a scalar wave equation as the generating function. One kind of vector wave function, called the Cartesian or rectilinear vector wave function, is formed if we let

F=×(Ψ1c)E91

where ψ1 denotes a characteristic function, which satisfies the scalar wave equation

2Ψ+κ2Ψ=0E92

And c denotes a constant vector, such as x, y, or z. For convenience, we shall designate c as the piloting vector and Ψ as the generating function. Another kind, designated as the spherical vector wavefunction, will be introduced later, whereby the piloting vector is identified as the spherical radial vector R.

Actually, substituting Eq. (91) into Eq. (92), it is

×[c(2Ψ1+κ2Ψ1)]=0E93

The set of functions so obtained

M1=×(Ψ1c)E94
N2=1κ××(Ψ2c)E95
L3=(Ψ3)E96

Ψ2,Ψ3 denote the characteristic functions which also satisfy (92) but may be different from the function used to define M1.

In the following, the expressions for the dyadic Green functions of a rectangular waveguide will be derived asserting to the vector wave functions. The method and the general procedure would apply equally well to other bodies (cylindrical waveguide, circular cylinder in free space, and inhomogeneous media and moving medium).

Figure 5 shows the orientation of the guide with respect to the rectangular coordinate system, and we will choose the unit vector z to represent the piloting vector c.

The scalar wave function

Ψ=(Acoskxx+Bsinkxx)(Ccoskyy+Dsinkyy)eihzE97

where kx2+ky2+h2=κ2.

the constants kx and ky should have the following characteristic values

kx=mπa,m=0,1,E98
ky=nπb,n=0,1,E99

The complete expression and the notation for the set of functions M, which satisfy the vector Dirichlet condition are

Memn(h)=×[Ψemnz]=(kyCxSyx+kxCySxy)eihzE100

where Sx=sinkxx,Cx=coskxx, Sy=sinkyy,Cy=coskyy. The subscript “e” attached to Memn is an abbreviation for the word “even,” and “o” for “odd.”

In a similar manner

Nomn=1κ(ihkxCxSyx+ihkyCySxy+(kx2+ky2)SxSyz)eihzE101

It is obvious that Memn represents the electric field of the TEmn mode, while Nomn represents that of the TMmn mode.

In summary, the vector wave functions, which can be used to represent the electromagnetic field inside a rectangular waveguide, are of the form

Me(o)mn=×[Ψe(o)mnz]E102
Ne(o)mn=1κ××[Ψe(o)mnz]E103

Then

G^m2(R,R)=+dhm,n(2δ0)κπab(kx2+ky2)[a(h)Nemn(h)Memn(h)+b(h)Momn(h)Nomn(h)]E104

where a(h)=b(h)=1κ2k2, h=±(k2kx2ky2)1/2and δ0={1,m=0orn=00,m0,n0.

M', N', m', n', h' denote another set of values, which may be distinct or the same as M, N, m, n, h.

## 6. Retarded and advanced Green functions

Green function is also utilized to solve the Schrödinger equation in quantum mechanics. Being completely equivalent to the Landauer scattering approach, the GF technique has the advantage that it calculates relevant transport quantities (e.g., transmission function) using effective numerical techniques. Besides, the Green function formalism is well adopted for atomic and molecular discrete-level systems and can be easily extended to include inelastic and many-body effects [13, 14].

(A) The definitions of propagators

The time-dependent Schrödinger equation is:

iħ|Ψ(t)t=H^|Ψ(t)E105

The solution of this equation at time t can be written in terms of the solution at time t′:

|Ψ(t)=U^(t,t)|Ψ(t)E106

where U^(t,t)is called the time-evolution operator.

For the case of a time-independent Hermitian Hamiltonian H^, so that the eigenstates |Ψn(t)=eiEnt/ħ|Ψnwith energies En are found from the stationary Schrödinger equation

H^|Ψn=En|ΨnE107

The eigenfunctions |Ψnare orthogonal and normalized, for discrete energy levels 1:

Ψm|Ψn=δmnE108

and form a complete set of states (I^is the unity operator)

nΨn|Ψn=1E109

The time-evolution operator for a time-independent Hamiltonian can be written as

U^(tt)=ei(tt)H^/ħE110

This formal solution is difficult to use directly in most cases, but one can obtain the useful eigenstate representation from it. From the identity U^=U^I^and (107), (109), (110) it follows that

U^(tt)=nei/ħEn(tt)|ΨnΨn|E111

which demonstrates the superposition principle. The wave function at time t is

|Ψ(t)=U^(t,t)|Ψ(t)=nei/ħEn(tt)Ψn|Ψ(t)|ΨnE112

where Ψn|Ψ(t)are the coefficients of the expansion of the initial function |Ψ(t)on the basis of eigenstates.

It is equivalent and more convenient to introduce two Green operators, also called propagators, retarded G^R(t,t)and advanced G^A(t,t):

G^R(t,t)=iħθ(tt)U^(t,t)=iħθ(tt)ei(tt)H^/ħE113
G^A(t,t)=iħθ(tt)U^(t,t)=iħθ(tt)ei(tt)H^/ħE114

so that at t > t′ one has

|Ψ(t)=iħG^R(tt)|Ψ(t)E115

while at t < t′ it follows

|Ψ(t)=iħG^A(tt)|Ψ(t)E116

The operators G^R(t,t)at t > t′and G^A(t,t)at t < t′ are the solutions of the equation

[iħtH^]G^R(A)(t,t)=I^δ(tt)E117

with the boundary conditions G^R(t,t)=0at t < t′, G^A(t,t)=0at t > t′. Indeed, at t > tEq. (118) satisfies the Schrödinger equation Eq. (105) due to Eq. (117). And integrating Eq. (117) from t=tηto t=t+ηwhere η is an infinitesimally small positive number η=0+, one gets

G^R(t+η,t)=1iħI^E118

giving correct boundary condition at t = t′. Thus, if the retarded Green operator G^R(t,t)is known, the time-dependent wave function at any initial condition is found (and makes many other useful things, as we will see below).

For a time-independent Hamiltonian, the Green function is a function of the time difference τ=t-t, and one can consider the Fourier transform

G^R(A)(E)=+G^R(A)(τ)eiEτ/ħdτE119

This transform, however, can not be performed in all cases, because G^R(A)(E)includes oscillating terms eiEτ/ħ. To avoid this problem we define the retarded Fourier transform

G^R(E)=limη0++G^R(τ)ei(E+iη)τ/ħdτE120

G^A(E)=limη0++G^A(τ)ei(Eiη)τ/ħdτE121

where the limit η → 0 is assumed in the end of calculation. With this addition, the integrals are convergent. This definition is equivalent to the definition of a retarded (advanced) function as a function of complex energy variable at the upper (lower) part of the complex plain.

Applying this transform to Eq. (117), the retarded Green operator is

G^R(E)=[(E+iη)I^H^]1E122

The advanced operator G^A(E)is related to the retarded one through

G^A(E)=G^R+(E)E123

Using the completeness propertyn|ΨnΨn|=1, there is

G^R(E)=n|ΨnΨn|(E+iη)I^H^E124

and

G^R(E)=n|ΨnΨn|E-En+iηE125

Apply the ordinary inverse Fourier transform to G^R(E), the retarded function becomes

G^R(τ)=-+G^R(E)eiEτ/ħdE2πħ=iħθ(τ)neiEnτ/ħ|ΨnΨn|E126

Indeed, a simple pole in the complex E plain is at E=Eniη, the residue in this point determines the integral at τ > 0 when the integration contour is closed through the lower half-plane, while at τ < 0 the integration should be closed through the upper half-plane and the integral is zero.

The formalism of retarded Green functions is quite general and can be applied to quantum systems in an arbitrary representation. For example, in the coordinate system Eq. (124) is

G^R(r,r,E)=nr|ΨnΨn|rE-En+iη=nΨn(r)Ψn(r)E-En+iηE127

(B) Path integral representation of the propagator

In the path integral representation, each path is assigned an amplitude eidtL, L is the Lagrangian function. The propagator is the sum of all the amplitudes associated with the paths connecting xa and xb (Figure 6). Such a summation is an infinite-dimensional integral.

The propagator satisfies

iG(xb,tb,xa,ta)=dxiG(xb,tb,x,t)iG(x,t,xa,ta)E128

Let us divide the time interval [ta, tb] into N equal segments, each of length Δt=(tbta)/N.

iG(xb,tb,xa,ta)=dx1dxNj=1NiG(xj,tj,xj1,tj1)=ANjdxjexp[iΔtL(tj,xj+xj12,xjxj12)]=D(x)eidtL(t,x,x˙)E129

where ln[iG(xj,tj,xj1,tj1)]=iΔtL(tj,xj+xj12,xjxj12).

Example: LC circuit-based metamaterials

In this section, we will use the relationship of current and voltage in the LC circuit to build the propagator of the LC circuit field coupled to an atom.

Figure 7 shows the LC-circuit.The following are valid:

I=dqdtE130
V=qC=LdIdtE131

Thus:

Cd2xdt2=xLE132

where x=LI, I is the current, V is the voltage, q is the charge quantity, L and C are the inductance and capacitance, respectively. Eq. (132) is equal to a harmonic, and the Lagrangian operator is:

L0(x,x˙)=12g(ε˙2ΩLC2ε2)E133

The Lagrangian operator describing the bipole is:

L0(x,x˙)=m2x˙2mΩ022x2E134

where x is the coordinate of the bipole, ε is the LC field, m is the mass of an electron, and e is the unit of charge. g=1c, and ΩLC=1LC. Defining their action items as:

SLC=dt[12g(ε˙2ΩLC2ε2)]E135

And

S0=dt[m2(x˙2Ω02x2)]E136

Taking the coupling effect (exε) into account, the Green function of the coupled system is:

G(x,ε)=DxDεeiSLC+iS0+idt[exε]E137

Where x represents the series coordinates x1,x2,…,and so on and ɛ represents ɛ1,ɛ2,…., and so on.

## 7. The recent applications of the Green function method

### 7.1. Convergence

In the Green function, the high oscillation of Bessel/Hankel functions in the integrands results in quite time-consuming integrations along the Sommerfeld integration paths (SIP) which ensures that the integrands can satisfy the radiation condition in the direction normal to the interface of a medium. To facilitate the evaluation, the method of moments (MoM) [15], the steepest descent path (SDP) method, and the discrete complex image method (DCIM) [16, 17] are very important methods.

The technique for locating the modes is quite necessary for accurately calculating the spatial Green functions of a layered medium. The path tracking algorithm can obtain all the modes for the configuration shown in Figure 8, even when region 2 is very thick [18]. Like the method in Ref. [19], it does not involve a contour integration and could be extended to more complicated configurations.

The discrete complex image method (DCIM) has been shown to deteriorate sharply for distances between source and observation points larger than a few wavelengths [20]. So, the total least squares algorithm (TLSA) is applied to the determination of the proper and improper poles of spectral domain multilayered Green’s functions that are closer to the branch point and to the determination of the residues at these poles [21].

The complex-plane for the determination of proper and improper poles is shown in Figure 9. Since half the ellipse is in the proper sheet of the -plane and half the ellipse is in the improper sheet, the poles will not only correctly capture the information of the proper poles but will also capture the information of those improper poles that are closer to the branch point kρ = k0.

For the 2-D dielectric photonic crystals as shown in Figure 10, the integral equation is written in terms of the unknown equivalent current sources flowing on the surfaces of the periodic 2-D cylinders. The method of moments is then employed to solve for the unknown current distributions. The required Green function of the problem is represented in terms of a finite summation of complex images. It is shown that when the field-point is far from the periodic sources, it is just sufficient to consider the contribution of the propagating poles in the structure [22]. This will result in a summation of plane waves that has an even smaller size compared with the conventional complex images Green function. This provides an analyzed method for the dielectric periodic structures.

Others, since the Gaussian function is an eigenfunction of the Hankel transform operator, for the microstrip structures, the spectral Green’s function can be expanded into a Gaussian series [23]. By introducing the mixed-form thin-stratified medium fast-multiple algorithm (MF-TSM-FMA), which includes the multipole expansion and the plane wave expansion in one multilevel tree, the different scales of interaction can be separated by the multilevel nature of the the fast multipole algorithm [24].

The vector wave functions, L, M, and N, are the solutions of the homogeneous vector Helmholtz equation. They can also be used for the analyses of the radiation in multilayer and this method avoids the finite integration in some cases.

### 7.2. Multilayer structure

The volume integral equation (VIE) can analyze electromagnetic radiation and scattering problems in inhomogeneous objects. By introducing an “impulse response” Green function, and invoking Green theorem, the Helmholtz equation can be cast into an equivalent volume integral equation including the source current or charges distribution. But the number of unknowns is typically large and the equation should be reformulated if there are in contrast both permittivity and permeability. At present, it is utilized to analyse the general scatterers in layered medium [25, 26].

When the inhomogeneity is one dimension, the Green function can be determined analytically in the spectral (Fourier) domain, and the spatial domain counterpart can be obtained by simply inverse Fourier transforming it.

Surface integral equation (SIE) method is another powerful method to handle electromagnetic problems. Similarly, by introducing the Green function, the Helmholtz equation can be cast into an equivalent surface integral equation, where the unknowns are pushed to the boundary of the scatterers [27].

Despite the convergence problem, the locations of the source and observation point may cause the change of Green function form, for example, for a source location either inside or outside the medium, the algebraic form of the Green functions changes as the receiver moves vertically in the direction of stratification from one layer to another [28].

First, we introduce the full-wave computational model [29]. A multilayer structure involving infinitely 1-D periodic chains of parallel circular cylinders in any given layer can be constructed as shown in Figure 11. Each layer consists of a homogeneous slab within which the circular cylinders are embedded. This is the typical aeronautic situation with fiber-reinforced four-layer pile (with fibers orientated at 0°, 45°, −45°, and 90°), but any other arrangement is manageable likewise.

In the multilayered photonic crystals, the Rayleig’s method and mode-matching are combined to produce scattering matrices. An S-matrix-based recursive matrix is developed for modeling electromagnetic scattering. Field expansions and the relationship between expansion coefficients are given.

There is a mix treatment for the inhomogeneous and homogeneous multilayered structure [30]. As shown in Figure 12, a substrate is divided into two regions. The top region is laterally inhomogeneous and for the finite-difference method (FDM) or the finite element method (FEM), the volume integral equation, is used. The bottom region is layerwise homogeneous, and the boundary-element methods (BEM) are used. The two regions are connected such as a BEM panel is associated with an FEM node on the interface.

A Green function was derived for a layerwise uniform substrate and was then used in a layerwise nonuniform substrate with additional boundary conditions applied to the interface. Given that the lateral inhomogeneity is local, volume meshing is used only for the local inhomogeneous regions, BEM meshing is applied to the surfaces of these local regions.

For a field (observation) point in the jth layer and a source point in the kth layer, the Green function has the form:

Gjku,l=Gjk,0u,l+m=0m+n0n=0cmnϕjku,labεkγmn×cosmπxfacosnπyfbcosmπxsacosnπysbE138

where the superscripts u and l indicate the upper and lower solutions, respectively, depending on whether the field point (or observation point) is above or below the source point. a and b are the substrate dimensions in the x- and y- directions, respectively, and more details can be found in Refs. [31, 32].

The electromagnetic field in a multilayer structure can be efficiently simplified by the assumption that the multilayer is grounded by a perfect electric conducto (PEC) plane [33, 34]. When the source and the field points are assumed to be inside the dielectric slab, in a layered medium as shown in Figure 13, by applying the boundary conditions, the 1-D Green functions is

Gx(x,x0;λx1,λx2)=(GxPMC+GxPEC)/2E139

where PMC represents the perfect magnetic conductor. The simplified Green function form can be deduced to the cae of (b).

The three-dimensional (3-D) Green function for a continuous, linearly stratified planar media, backed by a PEC ground plane, can also be expressed in terms of a single contour integral involving one-dimensional (1-D) green function. The constructure is shown in Figure 14.

The general formulation for a single electric current element has been worked out in detail in Ref. [35] which is based on the appropriate information from Ref. [36].

## How to cite and reference

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Jing Huang (June 14th 2017). Green Function, Recent Studies in Perturbation Theory, Dimo I. Uzunov, IntechOpen, DOI: 10.5772/68028. Available from:

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