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

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Chapter 10

Scalar Diﬀraction Theory

Diﬀraction is an important topic in the study of the propagation of electro-

magnetic waves. For a wave that is incident on an aperture in an opaque

screen, the propagation of the wave in front of the screen is called diﬀraction.

In fact the propagation of wave beams with ﬁnite transverse dimensions can

also be treated by means of the approach for diﬀraction problems. Diﬀraction

is a common phenomenon of wave propagation. The diﬀraction law based

on scalar theory had been established before J.C. Maxwell established his

systematic electromagnetic theory.

In this chapter we discuss only the scalar diﬀraction theory. This theory

is valid for cases in which the dimensions of the aperture are much larger

than the wavelength. The diﬀraction ﬁeld is dependent on the aperture form

and the incident waves. In general the incident waves include plane waves,

spherical waves and some kinds of beams.

We ﬁrst derive Fresnel–Kirchhoﬀ and Rayleigh–Sommerfeld diﬀraction

formulas from the scalar wave equation and Green theorem (in Section 10.1),

then deal with Fraunhofer diﬀraction and Fresnel diﬀraction for plane waves,

spherical waves, and Gaussian beams at round apertures (in Sections 10.2

and 10.3). Diﬀraction in anisotropic media is an important part of this

chapter, and we will discuss it in Sections 10.4 and 10.5. In the last section,

we deal with diﬀraction problems through an alternative approach, i.e. the

superposition of wave functions, and treat the propagation of wave beams in

isotropic, anisotropic, homogeneous, and inhomogeneous media.

10.1 Kirchhoﬀ’s Diﬀraction Theory

10.1.1 Kirchhoﬀ Integral Theorem

The basis of diﬀraction was founded by Huygens and Fresnel. Huygens pro-

posed the construction theory, Fresnel supplemented it with the interference

principle of the secondary wavelets, and the combination is the so-called

622 10. Scalar Diﬀraction Theory

Huygens–Fresnel principle. Later Kirchhoﬀ put it on a sound mathematical

basis by expressing the ﬁeld solution at any point in a closed surface as an in-

tegral in term of the ﬁeld amplitude and its ﬁrst derivative on the boundary,

which was realized through Green’s theorem. The second Green theorem is

expressed as

[ψ(x

)∇

G(x

, x) −G(x

, x)∇

ψ(x

)]dV

[ψ(x

)∇

G(x

, x) −G(x

, x)∇

ψ(x

)] · dS

, (10.1)

where V

is a volume bounded by a closed surface S

, the direction of dS

is along the outward normal from S, and x and x

are vector coordinates of

points inside and on the closed surface. In the integration, x

is a variable and

x is a constant. ψ is a scalar function which represents an electromagnetic

component and satisﬁes the scalar Helmholtz equation

∇

ψ + k

ψ = 0. (10.2)

G(x

, x) is the Green function of Helmholtz’s equation, which satisﬁes

(∇

+ k

)G(x

, x) = −

δ(x

− x)

. (10.3)

In Section 1.5, it was proved that the solution of (10.3) in a uniform

medium is a spherical wave emitted from a point source

G(x

, x) =

−jkr

4π²r

, (10.4)

where r = |x − x

|. Expression (10.4) is called as the Green function of

the Helmholtz equation in unbounded space. It is diﬀerent from the Green

function of Poisson’s equation by a factor e

−jkr

, i.e. a term of wave propa-

gation. Substitution of (10.3) and (10.4) into (10.1) yields the expression of

ﬁeld amplitude at an arbitrary point within the closed surface

ψ(x) = −

4π

ψ(x

)∇

−jkr

−

−jkr

∇

ψ(x

)

· dS

= −

4π

−jkr

·µ

jk +

ψ(x

) − ∇

ψ(x

)

· dS

, (10.5)

where the direction of dS

is taken as outward from the boundary. For

convenience, n is taken as the inward unit vector normal to the boundary,

so dS

= −ndS

, and (10.5) is rewritten as

ψ(x) = −

4π

−jkr

n ·

−

jk +

ψ(x

) + ∇

ψ(x

)

, (10.6)

10.1 Kirchhoﬀ’s Diﬀraction Theory 623

Figure 10.1: Diﬀraction from an aperture on a plane opaque screen.

which is called as the Kirchhoﬀ integral theorem. With it, the complex am-

plitude of electromagnetic ﬁeld can be derived from the complex amplitude

and its gradient on the boundary. In the integral, e

−jkr

/r denotes a spher-

ical wave from a point source, the amplitude of which is the dot product

between n and a vector function expressed in the square bracket. In other

words, the ﬁeld in a closed boundary is expressed as the interferential su-

perposition of waves emitted from point sources on the boundary, which are

the secondary wavelets proposed by Huygens. The interferential principle of

secondary wavelets was proposed by Fresnel, so (10.6) is the mathematical

representation of the Huygens–Fresnel principle.

10.1.2 Fresnel–Kirchhoﬀ Diﬀraction Formula

As shown in Fig. 10.1, a monochromatic wave is incident on a plane opaque

screen with an aperture. The incident wave may be of any distribution if the

variation of its phase and amplitude across the aperture is small and smooth.

P is a point at which the complex amplitude is to be determined. We denote

this point as the observation point or ﬁeld point. The dimensions of the

aperture are small compared to the distance between P and the screen, and

large compared to the wavelength.

The integrating domain of formula (10.6) can be divided into three parts

which are the aperture, the non-illuminated side of the opaque screen ex-

cluding the aperture and a portion of a spherical surface at inﬁnite distance.

The origin of the coordinate is located at an arbitrary point in the aperture.

x is the position vector of the observation point, x

is the position vector

624 10. Scalar Diﬀraction Theory

of a point on the boundary, and we assume that r

= |x|, r

= |x

|, and

r = |x − x

We ﬁrst carry out integration on the opaque screen and the aperture.

The value of ψ and ∇

ψ on them are never known exactly, and consequently

Kirchhoﬀ made the following assumptions.

1. In the aperture, ψ and its derivative along the normal from S, ∂ψ/∂n,

are identical to those of the incident wave in the absence of the opaque

screen.

2. On the opaque screen, ψ = 0 and ∂ψ/∂n = 0.

These assumptions are called Kirchhoﬀ’s boundary conditions and are the

basis of Kirchhoﬀ diﬀraction theory. Obviously, Kirchhoﬀ’s boundary condi-

tions do not agree with rigorous electromagnetic theory. As the dimensions

of the aperture are much larger than the wavelength, the ﬁrst assumption

will not introduce much error. Only the ﬁeld near the rim of the aperture

is disturbed, and as long as the aperture is large enough compared with

the wavelength, the edge eﬀect can be ignored. Nevertheless, the second as-

sumption is strictly contradictory to electromagnetic theory. According to

electromagnetic theory, if the ﬁeld amplitude on a surface and its derivative

along the normal to the surface are zero in the ﬁeld region, the ﬁeld will be

zero everywhere. Even so, if the dimensions of the aperture are much larger

than the wavelength, Kirchhoﬀ’s b oundary conditions are still acceptable.

The second assumption can be modiﬁed to avoid the contradiction through

choosing an appropriate Green function.

Next we analyze the integration on the spherical portion. Obviously, the

ﬁeld on the spherical surface is caused by the disturbance in the aperture.

As r

→ ∞, the ﬁeld amplitude on the spherical surface is

ψ(x

) =

−jkr

, (10.7)

and its derivative along the inward normal from S

n · ∇

ψ(x

) =

jk +

ψ(x

). (10.8)

Obviously, if r

→ ∞, then r/r = n, 1/r = 1/r

. On the spherical surface we

have

n ·

−

jk +

ψ(x

) + ∇

ψ(x

)

= 0. (10.9)

The integral on the spherical surface is zero.

By applying Kirchhoﬀ’s boundary conditions, we ﬁnd the complex am-

plitude at an arbitrary point in front of the aperture will be

ψ(x) = −

4π

−jkr

−

jk +

ψ(x

) cos α +

∂ψ(x

)

∂n

, (10.10)

10.1 Kirchhoﬀ’s Diﬀraction Theory 625

Figure 10.2: Right half-space Green functions of the ﬁrst kind (a), and the

second kind (b).

where S

is the aperture surface and α is the angle between r and n.

Kirchhoﬀ’s boundary conditions require that the observation point is far

from the aperture, otherwise the edge eﬀect may not be neglected. For r À λ,

(10.10) is simpliﬁed to

ψ(x) = −

4π

−jkr

−jkψ(x

) cos α +

∂ψ(x

)

∂n

, (10.11)

where the integrating domain is the aperture only.

10.1.3 Rayleigh–Sommerfeld Diﬀraction Formula

In the last subsection, the choice of the Green function of the unbounded

space leads to the requirement for the second item of Kirchhoﬀ’s boundary

conditions. To avoid the assumption that both the ﬁeld amplitude and its

derivative are zero on the opaque screen simultaneously, we can adopt Green

functions of half-space, which are divided into two kinds called the half-space

Green functions of the ﬁrst kind and that of the second kind.

As shown in Fig. 10.2(a) and (b), P is a point to the right of a screen,

and P

is the mirror image point of P . If two point sources with the same

amplitude are placed at P and P

, they form the Green function of half-space.

If the two point sources have reverse phases, they compose the half-space

Green function of the ﬁrst kind. If they have identical phases, they compose

the half-space Green function of the second kind.

The half-space Green function of the ﬁrst kind is

G =

−jkr

4π²r

−

−jkr

4π²r

. (10.12)

626 10. Scalar Diﬀraction Theory

In the right half-space it satisﬁes (10.3). On the opaque screen and in the

aperture

G = 0. (10.13)

Choice of the half-space Green function of the ﬁrst kind will leave out the

requirement that the ﬁeld derivative is zero on the opaque screen.

The half-space Green function of the second kind is

G =

−jkr

4π²r

−jkr

4π²r

. (10.14)

It satisﬁes (10.3) in the right half-space. On the opaque screen and in the

aperture

∇

G · n =

∂ G

∂ n

= 0. (10.15)

Choice of the half-space Green function of the second kind will leave out the

requirement that the ﬁeld amplitude is zero on the opaque screen.

On the screen and in the aperture r

= r

= r and the gradient of the

half-space Green function of the ﬁrst kind is

∇

G = (∇

− ∇

)

−jk −

−jkr

4π²r

= 2n cos α

+ jk

−jkr

4π²r

≈ 2jkn cos α

−jkr

4π²r

, (10.16)

where α is the angle between r

and the normal of the screen. Substitution

of (10.16) into (10.1) yields the diﬀraction ﬁeld

ψ(x) =

2π

−jkr

cos αψ(x

)dS

. (10.17)

In obtain (10.17), G = 0 in the aperture and r À λ are used.

From the half-space Green function of the second kind, we derive the

diﬀraction ﬁeld

ψ(x) = −

2π

−jkr

∂ψ(x

)

∂n

. (10.18)

The above formulas are two kinds of Rayleigh–Sommerfeld diﬀraction

formulas. In applying the half-space Green functions to derive the diﬀraction

ﬁeld, we can assume the ﬁeld distribution on the opaque screen separately.

For the half-space Green function of the ﬁrst kind, only the ﬁeld itself is

assumed to be zero; for the second sort of half-space Green function only

the derivative of the ﬁeld is assumed to be zero, and this has avoided the

contradiction in Kirchhoﬀ’s boundary condition.

From (10.11), (10.17), and (10.18) we know that the Fresnel–Kirchhoﬀ

diﬀraction formula is the average of two kinds of Rayleigh–Sommerfeld

diﬀraction formulas.

10.2 Fraunhofer and Fresnel Diﬀraction 627

Figure 10.3: Coordinate system for the diﬀraction of spherical wave.

10.2 Fraunhofer and Fresnel Diﬀraction

10.2.1 Diﬀraction Formulas for Spherical Waves

Fig. 10.3 shows that a spherical wave sent from Q is incident on an aperture

in a opaque screen. The coordinate system is set to make the screen lie in the

xy plane and the origin O be a point in the ap erture. P is the observation

point and M is an arbitrary point in the aperture. The angles which the lines

QM and P M make with the normal are β and α, respectively. The complex

amplitude of the incident wave in the aperture is

ψ(x

) =

−jkR

. (10.19)

The derivative of ψ along the p ositive z direction is

∂ψ

∂n

= −

jk +

cos βψ(x

) ≈ −jk cos βψ(x

). (10.20)

Substitution of (10.19) and (10.20) into (10.11), (10.17), and (10.18) yields

ψ(x) =

4π

(cos α + cos β)

−jk(r+R)

, (10.21)

ψ(x) =

2π

cos α

−jk(r+R)

, (10.22)

ψ(x) =

2π

cos β

−jk(r+R)

. (10.23)

It is meaningless to discuss which of the three diﬀraction formulas is more

accurate, because they are all derived under some approximate conditions.

628 10. Scalar Diﬀraction Theory

Near the edge of the aperture the assumed boundary conditions are much

more diﬀerent from the real ones. In order to reduce the edge eﬀect we need

to make some restrictions. First, the dimensions of the aperture are much

larger than the wavelength. Second, the distances of the observation point

and the source point from the aperture are much larger than the dimensions

of the aperture. Third, the phase variation of the incident wave across the

aperture is small and smooth, which means that the incident angle must not

be too large. Based on these restricting conditions, cos α and cos β are nearly

invariable over the aperture. Furthermore we may predict that the diﬀraction

ﬁeld must distribute near a line which passes through Q and the aperture

center, which means α ≈ β, and the three diﬀraction formulas are identical.

Now we investigate minutely the diﬀraction integral under the paraxial

condition. Letting cos α = cos β = 1 in (10.21), we obtain

ψ(x) =

2π

−jk(r+R)

, (10.24)

where R and r in the dominator can be replaced by R

, the distance of the

source point to the coordinate origin, and r

, the distance of the observation

point to the coordinate origin, respectively. R and r in the exponential factor

cannot be replaced, because they are related to the phase variation. Generally

r + R will change by many wavelengths as M is at diﬀerent locations. Then

(10.24) is simpliﬁed to

ψ(x) =

2πR

−jk(r+R)

. (10.25)

The distance from P (x, y, z) to M(x

, y

, 0) is

r =

(x − x

)

+ (y − y

)

+ z

≈ r

−

+ yy

+ y

, (10.26)

where r

+ y

+ z

. The distance from Q(x

, y

, z

) to M(x

, y

, 0) is

R = QM =

− x

)

+ (y

− y

)

+ z

≈ R

−

+ y

, (10.27)

where R

+ y

+ z

. Substituting (10.26) and (10.27) into (10.25)

yields

ψ(x) =

2πR

−jk(R

)

Z Z

jkf (x

)

, (10.28)

where

f(x

, y

−

. (10.29)

10.2 Fraunhofer and Fresnel Diﬀraction 629

Figure 10.4: The pattern of Fraunhofer diﬀraction.

If R

and r

are so large that the phase variation caused by the quadratic

terms of x

and y

in (10.29) can be neglected, the result of (10.28) is called

Fraunhofer diﬀraction. In fact it is the diﬀraction of a plane wave with the

observation point located at inﬁnite distance.

In practical applications it is not necessary to place the wave source and

the observation point at an inﬁnite distance. In Fig. 10.4, a lens parallel to

the screen is located to its left, and the point source is placed at the focal

plane of the lens. In this way the incident wave will be approximately a plane

wave. To the right of the screen, another lens parallel to the screen is placed,

and at the focal plane of the second lens the pattern of Fraunhofer diﬀraction

will be displayed on an observation screen.

If the quadratic term of (10.29) cannot be neglected, the result is called

Fresnel diﬀraction, which is the diﬀraction with the observation point at a

ﬁnite distance. Fraunhofer diﬀraction is suitable for calculations of the far-

ﬁeld distribution, and Fresnel diﬀraction is suitable for calculations of the

distribution in a medium range. For the near-ﬁeld, Kirchhoﬀ’s boundary

conditions are not valid, and the diﬀraction theory is not suitable for calcu-

lations of this kind of ﬁeld distribution.

10.2.2 Fraunhofer Diﬀraction at Circular Apertures

As mentioned previously, Fraunhofer diﬀraction is the diﬀraction of plane

waves with the observation point at an inﬁnite distance. For a normally

incident plane wave both x

and y

are zero in (10.29). Neglecting

the second-order term of x

and y

in (10.29) and substituting it into (10.28),

630 10. Scalar Diﬀraction Theory

Figure 10.5: Coordinate system for the Fraunhofer diﬀraction at a circular

aperture.

we obtain the ﬁeld amplitude of Fraunhofer diﬀraction:

ψ =

Z Z

exp

¶¸

, (10.30)

where A is a constant. As shown in Fig. 10.5, we can carry out the integral

in a polar coordinate system of ρ

and φ

. For the diﬀraction at a circular

aperture with radius a the origin of the coordinate system is at the center of

the aperture.

By coordinate substitution that x

= ρ

cos φ

and y

= ρ

sin φ

, (10.30)

becomes

ψ =

dρ

2π

exp

cos φ

sin φ

¶¸

dφ

. (10.31)

From Fig. 10.5, we have

= sin θ cos γ,

= sin θ sin γ. (10.32)

Substitution of (10.32) into (10.31) yields

ψ =

dρ

2π

exp[jkρ

sin θ cos(φ

− γ)]dφ

. (10.33)

Since γ is invariant, (10.33) is the same as

ψ =

dρ

2π

exp(jkρ

sin θ cos φ

)dφ

. (10.34)