Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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8.11 Magnetostatic Waves 571

Figure 8.38: The dispersion curve of MSSW.

i.e.,

Ω

+ Ω

≤ Ω ≤ Ω

, (8.461)

γµ

+ M

) ≤ ω ≤ γµ

, (8.462)

and we have

β = 0 when Ω =

Ω

+ Ω

, and β → ∞ when Ω = Ω

We can tell from Fig. 8.34 that 1 + χ

≥ 0 when Ω ≥

Ω

+ Ω

. This is

the frequency range for the existence of MSSW.

When the operating frequency is within the range given in (8.462), there

is only one root, which corresponds to the MSSW modes, satisfying the eigen-

value equation (8.457). The Ω-β diagram for MSSW is plotted in Fig. 8.38

according to (8.457). We can ﬁnd from this diagram that the slope of the Ω

vs. β curve is positive. Hence MSSW is a forward wave, for which the group

velocity and the phase velocity are in the same direction.

Film. In the case of k

6= 0 and k

6= 0, the eigenvalue equation remains

(8.442) and the wave vector of a MSW in an arbitrary direction along the

ﬁlm is given by

k =

kβ =

, β =

+ k

The dispersion surfaces, i.e., the curved surfaces determined by the eigenvalue

equation (8.442) plotted in the Ω-k

-k

space are shown in Fig. 8.39.

It is seen that as k

approaches zero, the frequency range of the MSSW

decreases to zero and only the MSBVW propagates along z. The dispersion

572 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.39: Dispersion surfaces of MSW in ferrimagnetic ﬁlm with d-c mag-

netic ﬁeld parallel to the ﬁlm.

curves are the same as those shown in Fig. 8.35. On the other hand, as k

approaches zero, the frequency range of the MSBVW vanishes and only the

MSSW propagates along y. The dispersion curve is the same as that shown

in Fig. 8.38. The MSBVW and MSSW can coexist only when the direction

of propagation makes an arbitrary angle other than 0 and π/2 with respect

to the direction of the d-c biasing ﬁeld.

(2) D-C Magnetic Field Perpendicular to the Film Surface,

MSFVW

Suppose that a ferrimagnetic ﬁlm or slab of thickness s lies on the x-y plane,

the two surfaces of the slab are at z = ±s/2, and the ﬁlm is unbounded in

the x and y directions. The d-c magnetic bias ﬁeld is in the z direction and

is suﬃciently strong for the magnetization of the ferrimagnetic ﬁlm to be

saturated. See Fig. 8.40.

The magnetic potentials inside and outside the ferrimagnetic ﬁlm are ϕ

and ϕ

, respectively,

(x, y, z) = X

(x)Y

(y)Z

(z), |z| ≤ s/2, (8.463)

(x, y, z) = X

(x)Y

(y)Z

(z), |z| ≥ s/2. (8.464)

In order to satisfy the b oundary conditions at z = ±s/2, we must have

(x) = X

(x) = X(x) and Y

(y) = Y

(y) = Y (y). We consider the traveling

waves along +x and +y only, These functions may then have the form

X(x) = e

−jk

, Y (y) = e

−jk

. (8.465)

8.11 Magnetostatic Waves 573

Figure 8.40: MSW in ferrimagnetic ﬁlm with a d-c magnetic ﬁeld perpendic-

ular to the ﬁlm.

Inside the ferrimagnetic slab, function Z

(z) must be in the form of standing

wave, i.e.,

(z) = A sin k

z + B cos k

z, |z| ≤

. (8.466)

Outside the ferrimagnetic ﬁlm, to satisfy the natural boundary conditions at

z → ±∞, function Z

(z) must be of the form of decaying ﬁelds, i.e.,

(z) = Ce

−τ

, z ≥

, (8.467)

(z) = De

, z ≤ −

. (8.468)

The magnetic potentials inside and outside the ferrimagnetic slab take the

following forms:

(x, y, z)=X(x)Y (y)Z

(z) = (A sin k

z +B cos k

z) e

−jk

, |z| ≤

(8.469)

(x, y, z) = X(x)Y (y)Z

(z) = Ce

−τ

−jk

, z ≥

, (8.470)

(x, y, z) = X(x)Y (y)Z

(z) = De

−jk

, z ≤ −

. (8.471)

Substituting them into (8.418) and (8.419) yields

(1 + χ

)

+ k

= 0, → k

= −(1 + χ

)

+ k

, (8.472)

+ k

− τ

= 0, → τ

= k

+ k

. (8.473)

The boundary conditions at z = ±s/2 are

z=±s/2

= H

z=±s/2

→ ϕ

z=±s/2

= ϕ

z=±s/2

, (8.474)

z=±s/2

= B

z=±s/2

→

∂ϕ

∂z

z=±s/2

∂ϕ

∂z

z=±s/2

. (8.475)

574 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

Figure 8.41: The dispersion curves of MSFVW.

Using (8.469)–(8.471) in the boundary equation (8.474) and (8.475) at

+s/2 and −s/2 gives

A sin(k

s/2) + B cos(k

s/2) = Ce

−τ

s/2

, (8.476)

−A sin(k

s/2) + B cos(k

s/2) = De

−τ

s/2

, (8.477)

cos(k

s/2) − Bk

sin(k

s/2) = −τ

−τ

s/2

, (8.478)

cos(k

s/2) + Bk

sin(k

s/2) = τ

−τ

s/2

. (8.479)

From these four equations we obtain

tan

. (8.480)

Using (8.472) and (8.473) in the above equation yields

tan

−(1 + χ

)

+ k

−1

1 + χ

, i.e., tan

βs

= K, (8.481)

where, K =

−1

1 + χ

and β =

+ k

is the angular wave number of

the wave propagating in an arbitrary direction along the ﬁlm. This is the

eigenvalue equation for the magnetostatic wave in a ferrimagnetic ﬁlm with

a d-c magnetic ﬁeld perpendicular to the ﬁlm surface. We ﬁnd that the

propagation characteristics are independent of the direction of propagation

along the ﬁlm.

The condition for the solution of β for this equation to exist in the real

domain is also K

> 0:

Ω

≤ Ω ≤

Ω

+ Ω

, i.e., γµ

≤ ω ≤ γµ

+ M

), (8.482)

Problems 575

which is the same as it is for MSBVW. Following (8.481) we work out that

β = 0 when Ω = Ω

, and β → ∞ when Ω =

Ω

+ Ω

The Ω-β diagram is plotted as shown in Fig. 8.41 according to (8.481).

This diagram shows that the slopes of the Ω vs. β curves are positive. Hence

these waves are forward waves.

Within the frequency range for MSW propagation, k

and τ

are real.

The ﬁeld distributions in the z direction inside the ferrimagnetic ﬁlm are

in standing-wave form and those outside the ﬁlm are decaying ﬁelds. This

means that this kind of wave is a volume wave or bulk wave.

Now we conclude that in a ferrimagnetic ﬁlm with a d-c magnetic ﬁeld

perpendicular to the ﬁlm, the magnetostatic wave along the ﬁlm is a magne-

tostatic forward volume wave denoted by MSFVW. The distributions of ϕ

and ϕ

with resp ect to z for MSFVW are similar to those for MSBVW, given

in Fig. 8.36.

The space-charge wave is a type of non-maxwell wave caused by the

electric-ﬁeld-charge interaction, and the magnetostatic wave is a type of non-

maxwell wave caused by the magnetic-ﬁeld-spin interaction.

Problems

8.1 In plasma, the d-c electric ﬁeld is neglected and the potential is supposed

to be uniform throughout the plasma. The ions are at least 1840 times

as heavy as the electrons and will be considered immobile.

Assume a neutral plasma in which all electrons in the region x to x+∆x

are given a displacement dx. Considering the force acting to restore

neutrality, show that the displaced sheet of electrons will oscillate at

the plasma frequency ω

given by (8.33).

8.2 Prove that ²

(ω) and ²

(ω) given in (8.13) and (8.14) satisfy the Kramers–

Kronig relations.

8.3 Show that the ratio of magnetic to electric forces on the electrons re-

sulting from the time-varying ﬁelds of a uniform plane wave is v

where v

is the electron velocity and v

is the phase velocity of the

wave. Hence the time-varying magnetic force on the electron may be

neglected except that the speed of electron is close to the speed of light.

8.4 Find the energy velocity of a plane wave in a dispersive medium using

the expression of electric ﬁeld stored energy density (1.203). Show that

it is the same as (8.62).

8.5

Show that at the low-frequency end, ω ¿ ω

, the refractive index of

dielectric material is real and is independent of frequency. Use the

ideal gas model.

576 8. Electromagnetic Waves in Dispersive Media and Anisotropic Media

8.6 Find the phase velocity of dielectric material at the low-frequency end,

ω ¿ ω

. Show that, in this case, the phase velocity approaches a

constant value less than c. Use the ideal gas model.

8.7 Find the group velocity and the energy velocity of dielectric material at

the low-frequency end, ω ¿ ω

. Show that, in this case, the group

velocity and the energy velocity approach the phase velocity. Use the

ideal gas model.

8.8 Show that, at the high-frequency end, ω À ω

, the phase velocity, the

group velocity, and the energy velocity approach c. Use the ideal gas

model.

8.9 Show that the eigenvalue equations for plane waves in reciprocal media,

(8.181) and (8.182), can be derived from the Fresnel normal equation

(8.183).

8.10 Sketch the index ellipsoid along with index surface for barium titanate

(BaTiO

) with ²

= 5.94 and ²

= 5.59.

8.11 A light ray with an arbitrarily oriented D is incident on a planar uni-

axial crystal slab. The ray will split into an ordinary ray and an ex-

traordinary ray. Show that these two rays are parallel to each other

when they pass through the slab.

8.12 A linearly polarized wave with its wave vector in the x direction is

incident on a uniaxial crystal with the optical axis in the z direction.

The electric ﬁeld vector lies on the y-z plane and the angle it makes

with the z axis is 45

◦

. Find the distance it takes for the propagating

wave to transform the wave into a circularly polarized wave.

8.13 Find the gyromagnetic ratio by classical theory. Suppose that the elec-

tron can be viewed as a uniform mass and uniform charge distribution

in a spherical volume. Prove that the result has an error of a factor of

2 compared with the result found by quantum theory.

8.14 If a linearly polarized wave is incident on a magnetized ferrite, ﬁnd

the condition under which the wave can be transformed into circularly

polarized wave.

8.15 Find the eigenvalue equations for two eigenwaves in magnetized plasma

propagating in an arbitrary direction. Use kDB coordinates.

8.16 Find the eigenvalue equations and the angular wave numbers for a

MSSW propagating in a ferrimagnetic slab with a metallic coating on

one side. Show that the angular wave numbers for waves propagating

in opposite directions are diﬀerent.

Chapter 9

Gaussian Beams

Gaussian beams [38, 116] are important spatial distributions of electromag-

netic waves whose transverse amplitude distributions are some kind of Gaus-

sian functions. The optical distributions of some laser beams and the modal

ﬁeld distributions of optical waveguides with special index proﬁles may be

Gaussian. Gaussian beams have been found to have attentions with the ap-

pearance of lasers, and the development of lasers and optoelectronics make

them more and more important.

Gaussian beams are not the exact solutions of electromagnetic ﬁeld equa-

tions, they are only the approximate solutions under some deﬁnite condi-

tions. In many situations they are accurate enough, especially in the case

where the width of the beam waist is much larger than the wavelength. In

this chapter, we will derive the distributions of a variety of Gaussian beams

in homogeneous, quadratic index, and anisotropic media, and discuss their

characteristics.

We pay more attention to the transformation of Gaussian beams through

optical systems, and to do this we introduce the q parameter and so-called

ABCD law, which is widely used in geometric optics.

9.1 Fundamental Gaussian Beams

From electromagnetic ﬁeld theory, as a wave is conﬁned in a relatively small

volume where the dimension of the volume is comparable with the wave-

length, the vector wave equation must be used to derive the ﬁeld distribu-

tion. In chaps. 4–7 we discussed this kind of problem. In unbounded media

or in free space, the uniform plane wave is the simplest solution for the ﬁeld

distribution, and the superposition of elementary plane waves propagating in

diﬀerent directions can make up any desired ﬁeld distribution. In a paraxial

approximation, all elementary plane waves propagate nearly along the same

direction, i.e. the z axis. The electric and magnetic ﬁelds of these plane

578 9. Gaussian Beams

waves are in the directions nearly perpendicular to the axis. For such a ﬁeld

distribution we need to solve only a scalar wave equation in which the scalar

function may be a component of the electric or magnetic ﬁelds, or a compo-

nent of the vector potential. In many cases, denoting it as a component of

the vector potential will bring some convenience.

The scalar Helmholtz equation is

∇

ψ + k

ψ = 0. (9.1)

In (9.1), k

= k

= ω

, where n is the refractive index. Under the

paraxial condition, the solution of (9.1) can be taken as

ψ = u(x, y, z)e

−jkz

, (9.2)

where u(x, y, z) is a slowly varying function of z, which satisﬁes the conditions

∂u

∂z

¿ |ku|,

∂

∂z

∂u

∂z

¿ |k

u|. (9.3)

Substituting (9.2) into (9.1), we obtain the paraxial wave equation

∇

u(x, y, z) − 2jk

∂u

∂z

= 0, (9.4)

where ∇

= ∇

− ∂

/∂z

, which is the transverse Laplacian operator.

The fundamental mode of Gaussian beams, which is generally called the

Gaussian beam, is one of the solutions of the paraxial wave equation. Because

there is no boundary condition, the ﬁeld solution cannot obtained from (9.4)

directly, so the form of the ﬁeld distribution must be speciﬁed.

To make the transverse ﬁeld distribution Gaussian, in cylindrical coordi-

nate system with axial symmetry we take u in the form of

u = A exp

−j

p(z) +

kρ

2q(z)

¸¾

, (9.5)

where p(z) and q(z) are both complex functions. This assumption is based

on the following considerations.

1. The ﬁeld amplitude distribution at planes normal to the propagation

direction must be Gaussian.

2. The phase front must be relative to ρ

and z.

3. On the beam axis, the amplitude is a function of z, and the phase factor

is not a linear function of z.

The ﬁrst two items lead to kρ

/q(z) in (9.5) and the third leads to p(z).

9.1 Fundamental Gaussian Beams 579

Substituting (9.5) into (9.4), we obtain

2jk

+ k

+ 2k

= 0. (9.6)

The condition that (9.6) is valid for any value of ρ is that the coeﬃcients

of all orders of ρ must be zero, and this leads to two equations

= 0, and

= 0. (9.7)

The solutions of the above equations are

q = z + q

, and p = −j ln(z + q

). (9.8)

Substitution of p and q into (9.5) yields

u = A exp

−ln(z + q

) − j

kρ

2(z + q

)

, (9.9)

where A is a constant. To obtain the transverse Gaussian distribution, q

must be a complex number

= −z

+ js, (9.10)

where z

and s are both real numbers. Substituting (9.10) into (9.9), we

obtain

u =

(z − z

) + js

exp

(

−ksρ

(z − z

)

+ s

− j

k(z − z

)ρ

(z − z

)

+ s

)

. (9.11)

Introducing the normalization condition

∗

2πρdρ = 1, we obtain

A = j

. (9.12)

In (9.11), making the substitutions

(z)

(z − z

)

+ s

, (9.13)

R(z)

z − z

(z − z

)

+ s

, (9.14)

tan φ =

z − z

, (9.15)

we obtain

u =

exp

−ρ

exp

−j

kρ

− φ

¶¸

. (9.16)

580 9. Gaussian Beams

Substitution of (9.16) into (9.2) yields

ψ =

exp

−ρ

exp

−j

z +

− φ

¸¾

. (9.17)

In (9.16) and (9.17), w(z) is the beam radius, which is the distance from the

beam axis to where the ﬁeld amplitude is down to 1/e of its value on the

axis. At z = z

, w takes a minimum value, and this position is called the

beam waist. The radius of the beam waist is expressed as

= w(z

) =

. (9.18)

is a basic parameter of the Gaussian beam. The spatial distribution

of a Gaussian beam is determined totally by w

, its position z

, and the

wavelength λ. In terms of them, the beam parameters are expressed as

s =

nπw

, (9.19)

w(z) = w

1 +

λ(z − z

)

nπw

= w

1 +

z − z

, (9.20)

R(z) =

z − z

(z − z

)

nπw

z − z

(z − z

)

+ s

. (9.21)

The parameter s is called the confocal parameter, and the parameter R is the

curvature radius of the phase front, which will be further discussed later.

9.2 Characteristics of Gaussian Beams

9.2.1 Condition of Paraxial Approximation

Because the solution of the Gaussian beam is derived from the paraxial wave

equation, the ﬁeld distribution must be restricted by the paraxial approxi-

mation conditions (9.3). From (9.9) we obtain

∂u

∂z

= −

(z − z

) + js

− j

kρ

2[(z − z

) + js]

u. (9.22)

Substitution of (9.22) into (9.3) yields the paraxial condition

¿ k,

(z − z

)

+ s

¿ 1. (9.23)

Because most of the energy in the beam is conﬁned in the region of ρ < w,

ρ can be replaced by w in the second formula of (9.23). Thus the above two

conditions are identical. The paraxial condition is then

√

2nπ

. (9.24)