Kramers–Kronig Equations | 12 | Magnetics, Dielectrics, and Wave Propa

ABSTRACT

It is clear that we can represent the microwave properties of materials either by a permittivity (e) or permeability (m) constant. Both e and mmay be scalar or tensor quantities that are generally complex. The real part is associated with the dispersive properties while the imaginary part is associated with the absorptive properties of materials. In 1927, Kramers-Kronig (see ‘‘References’’) showed that the real part of the susceptibility is related to the imaginary part and vice versa. Let’s consider the time response of a magnetic system and, in particular,

the time dependence of the magnetic ﬁeld H(t). H(t) may be related to H(v) via the Fourier transform

H(t) ¼ ð1 1

H(v)ejvtdv (7:1)

and

H(v) ¼ 1 2p

H(t)ejvtdt: (7:2)

Since H(t) is a measurable quantity,

H(t) ¼ H*(t), (7:3)

which implies

H(t) ¼ ð1 1

H(v)ejvtdv ¼ ð1 1

H(v)ejvtdv

2 4

3 5*¼ ð1

1 H*(v)ejvtdv: (7:4)

Let z¼v

H*(v)ejvtdv ¼ ð1 1

H*(z)ejztdz ¼ ð1 1

H*(z)ejztdz: (7:5)

Let v¼ z

H*(z)ejztdz ¼ ð1 1

H*(v)ejvtdv ð1 1

H*(v)ejvtdv ¼ ð1 1

H(v)ejvtdv:

(7:6)

The two underlined integrals in Equation 7.6 are equal only if

H*(v) ¼ H(v): (7:7)

The magnetic ﬁeld is related to the magnetic ﬂux density, B(v), by

B(v) ¼ m(v)H(v), (7:8)

and

B*(v) ¼ m*(v)H*(v): (7:9)

Since B(v) is also a measurable quantity, we require that

B(v) ¼ B*(v), (7:10)

which implies

m(v)H(v) ¼ m*(v)H*(v): (7:11)

Use the property

H(v) ¼ H*(v), (7:12)

so that

[m(v) m*(v)]H(v) ¼ 0, (7:13)

where (in MKS)

m(v) ¼ 1þ x(v) m*(v) ¼ 1þ x*(v): (7:14)

Equation 7.13 also implies that

[x(v) x*(v)]H(v) ¼ 0: (7:15)

with

Since H(v) is a measurable quantity, i.e., H(v) 6¼ 0, we have

x(v) ¼ x*(v), (7:16)

where, for example,

x(v) ¼ x0(v) jx00(v): (7:17)

Substituting the above relation into Equation 7.16, we have two subsequent relations

x0(v) jx00(v) ¼ x0(v)þ jx00(v): (7:18)

resulting in

x0(v) ¼ x0(v), even function of v x00(v) ¼ x00(v), odd function of v: (7:19)

Let’s relate x0(v) to x00(v) by considering the integral

I ¼ þ C

vx(v)dv v2 v20

: (7:20)

The contour of integration is deﬁned in Figure 7.1 to avoid residues. Near the mathematical poles, use the following transformation (Figure 7.2):

v v0 ¼ reju: (7:21)

Thus,

dv ¼ rejujdu ¼ (v v0)jdu: (7:22)

Similarly, near the v0 pole, dv ¼ rejuj du ¼ (vþ v0)j du: (7:23)

The integral becomes

I ¼ P ð1 1

vx(v)dv v2 v20

þ ð0 p

v0x(v0)(v v0)j du 2v0(v v0) þ

v0x(v0)(vþ v0)j du 2v0(vþ v0) ¼ 0,

(7:24)

where we’ve used the approximation

v2 v20 ¼ (vþ v0)(v v0)jvﬃv0ﬃ 2v0(vv0), and v2 v20 ¼ (vþ v0)(v v0)jvﬃv0ﬃ 2v0(vþ v0):

(7:25)

Thus, the integral simpliﬁes to the following

I ¼ P ð1 1

vx(v)dv v2 v20

pj 2 [x(v0)þ x(v0)] ¼ 0, (7:26)

yielding

P ð1 1

vx(v)dv v2 v20

¼ pj 2

2x0(v0)½ : (7:27)

Using Equation 7.19,

P ð1 1

v(x0(v) jx00(v))dv v2 v20

¼ pjx0(v0): (7:28)

Since x0(v) is an even function of v (see Equation 7.19) and vx0(v) is odd

vx0(v)dv v2 v20

¼ 0: (7:29)

with

However, x00(v) is an odd function of v (see Equation 7.19) and vx00(v) is even

jvx00(v)dv v2 v20

¼ 2j ð1 0

vx00(v)dv v2 v20

: (7:30)

Combining above result with equation (7.28) we obtain

x0(v0) ¼ 2 p P ð1 0

vx00(v)dv v2 v20

: (7:31)

There is an inverse relation for x00(v). For this relation, we need to consider the integral

I ¼ þ C

x(v)dv v2 v20

: (7:32)

After applying the same integration contour procedure as before, we obtain

I ¼ P ð1 1

x(v)dv v2 v20

pj 2

x(v0) v0

¼ 0, (7:33)

yielding the result that

P ð1 1

x(v)dv v2 v20

¼ px 00(v0) v0

: (7:34)

Since x0(v) is an even function of v (see Equation 7.19)

x0(v)dv v2 v20

¼ 2 ð1 0

x0(v)dv v2 v20

: (7:35)

However, x00(v) is an odd function of v (see Equation 7.19)

x00(v)dv v2 v20

¼ 0: (7:36)

Finally, we come to the conclusion that

x00(v0) ¼ 2v0 p

P ð1 1

x0(v)dv v2 v20

: (7:37)

Equations 7.31 and 7.37 are referred to as the Kramers-Kronig relations. These relationships are valid for diagonal as well as off-diagonal elements of the susceptibility tensor. The mathematical procedure to calculate the susceptibility is to vary v0

over the frequency range of interest. Usually x0(v) or x00(v) is measured. Thus, the other quantity may be determined using Kramers-Kronig relations. MATLAB1 code is included in the appendix to calculate susceptibilities.