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

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10.2 Fraunhofer and Fresnel Diﬀraction 631

0 2 4 6 8

0.0

1.0

k a sin

Figure 10.6: Intensity distribution of the Fraunhofer diﬀraction at a circular

aperture.

By introducing the integral representation and the derivative formula of

Bessel function, we express (10.34) as

ψ =

2πA

(kρ

sin θ)dρ

πa

(ka sin θ)

ka sin θ

, (10.35)

where J

and J

are Bessel functions of the zeroth order and the ﬁrst order.

The diﬀraction intensity is then

I =

(ka sin θ)

ka sin θ

, (10.36)

where I

is a constant, I

≈ I

is the intensity on the axis. The

intensity distribution is shown in Figure 10.6.

The intensity distribution is dependent on the diﬀraction angle θ. At

θ = 0 it has the principal maximum, and with the increase in θ it oscillates

with gradually diminishing amplitude. When the intensity is zero, there are

concentric dark rings. The dark rings correspond to the roots of the ﬁrst-

order Bessel function; that is

(ka sin θ) = 0. (10.37)

The ﬁrst dark ring occurs where sin(θ) = 3.832λ/2πa = 1.22λ/d, where d is

the diameter of the aperture. It is easy to prove that about 84% of the total

energy is contained in the region bounded by the ﬁrst dark ring, and about

90% of the total energy is in the region bounded by the second dark ring.

Where the intensity takes the maxima, there are concentric bright rings. The

diﬀraction angles corresponding to the bright rings are determined by

(ka sin θ) = 0, (10.38)

632 10. Scalar Diﬀraction Theory

Figure 10.7: Coordinate system for the Fresnel diﬀraction at a circular aper-

ture.

but the principal maximum of the intensity is located on the axis.

10.2.3 Fresnel Diﬀraction at Circular Apertures

In Fraunhofer diﬀraction, the source and the observation point are both at

an inﬁnite distance from the aperture, but in Fresnel diﬀraction at least the

observation point is at a ﬁnite distance. The calculation of Fresnel diﬀraction

is very complicated compared with that of Fraunhofer diﬀraction. Even for

a long narrow slit it will still involve the Fresnel integrals. With the applica-

tion of electronic computers this problem becomes simple, and the classical

method is unnecessary. In this section we will discuss only the simplest ex-

ample to illustrate the basic properties of Fresnel diﬀraction.

We deal with Fresnel diﬀraction of a spherical wave normally incident on

a circular aperture, as shown in Figure 10.7. For simplicity, we calculate

only the diﬀraction ﬁeld distribution on the axis passing through the center

of the aperture. The calculation is carried out in a polar coordinate system.

Expression (10.24) is directly used to derive the diﬀraction ﬁeld distribution.

In the polar coordinate system the integral is expressed as

ψ = jk

exp[−jk(r + R)]ρ

dρ

. (10.39)

From the relations ρ

+ r

= r

and ρ

+ R

= R

, we obtain

d(r + R) =

dρ

. (10.40)

10.2 Fraunhofer and Fresnel Diﬀraction 633

Substitution of (10.40) into (10.39) yields

ψ = jk

r + R

exp[−jk(r + R)]d(r + R). (10.41)

For the case that r

and R

are much larger than the radius of the aperture,

(10.41) becomes

ψ =

+ R

exp(−jkξ)dξ =

+ R

−jkξ

− e

−jkξ

, (10.42)

where ξ

= r

+ R

, ξ

+ r

+ R

. Expression (10.42) can be

further expressed as

ψ =

+ R

exp[−jk(r

+ R

)]

1 − exp

−jk

+ r

+ R

− r

− R

¶¸¾

= ψ

(1 − e

−jkq

), (10.43)

where ψ

is the amplitude at P of a spherical wave sent from a source located

at Q in the absence of the opaque screen. q is the route diﬀerence between

QMP and QP where M is a point at the rim of the circular aperture. The

route diﬀerence q is expressed as

q =

+ r

+ R

− (r

+ R

). (10.44)

From (10.43), the optical intensity at P is

I = 4I

sin

= 4I

sin

πq

, (10.45)

where I

is the intensity in the absent of the opaque screen. If q is a multiple

of half-wavelength, the optical intensity is maximum or zero. Accordingly,

the radius of the aperture satisﬁes

+ r

+ R

− (r

+ R

) =

nλ

, (10.46)

that is,

P M + QM − P Q =

nλ

, (10.47)

where n is an integer. Under the condition formulated by (10.47), the aper-

ture is said to contain n half-period zones. At the observation point P ,

the ﬁeld amplitudes caused by two neighboring half zones cancel each other.

Whether the intensity is maximum or minimum depends on whether n is an

even number or an odd number. If n is an even number, the intensity is

634 10. Scalar Diﬀraction Theory

Figure 10.8: Fresnel half-zone lens.

zero. If n is an odd number, the intensity is maximum. In Fig. 10.8, the

alternate half zones are blackened, and the amplitude at p is N times that

caused by a half zone; here N is the total number of the transparent half

zones. Such a zone plate is called a Fresnel half-zone lens, which is widely

used in optical instruments. If the alternate half zones are not blackened,

and instead an additional phase shift of half wavelength is attached to them,

the optical amplitude will be M times that caused by a half zone; here M is

the total number of half zones.

10.3 Diﬀraction of Gaussian Beams

Because of the importance of Gaussian beams in modern optics and opto-

electronics, it is necessary to study their diﬀraction [108]. In this section

we discuss only the diﬀraction at circular apertures. For apertures of other

forms, there are no analytical solutions in general.

10.3.1 Fraunhofer Diﬀraction of Gaussian Beams

In Fraunhofer diﬀraction, the incident wave front at the aperture is required

to be planar, and this condition can be satisﬁed only as the aperture is located

at the beam waist or far from the beam waist. The latter is the same as the

diﬀraction of a plane wave, and in this section we discuss only the former.

As shown in Fig. 10.9, a Gaussian beam is normally incident on a circular

aperture whose radius is a. The beam waist is located at the aperture, and its

10.3 Diﬀraction of Gaussian Beams 635

Figure 10.9: Fraunhofer diﬀraction of Gaussian beam at a circular aperture

located at the beam waist.

radius is larger than that of the aperture, otherwise the aperture is unneces-

sary. On the other hand, if the radius of the aperture is too small compared

with the beam waist, the situation will be identical to the diﬀraction of a

plane wave. With the same approach as in dealing with the diﬀraction of

a plane wave, the integral formula of Fraunhofer diﬀraction of a Gaussian

beam at a circular aperture is expressed as

ψ =

exp

−

(kρ

sin θ)ρ

dρ

, (10.48)

where A is a constant, w

is the radius of the incident beam waist, θ is the

diﬀraction angle, and r

is the distance from the observation point to the

center of the aperture. Under the condition that a

¿ w

the exponential

term in the integral is expanded as

exp

−

= 1 −

− ···. (10.49)

Here we introduce the series expansion of Bessel function of the zeroth order:

2ρ

= 1 −

− ···. (10.50)

Obviously, if a

¿ w

, replacing the exponential term in the integral with

the Bessel function is better than neglecting the second-order term in (10.49),

and we make such a replacement. The integral is then

ψ =

2ρ

(kρ

sin θ)ρ

dρ

. (10.51)

According to the integral formula for the product of Bessel functions, the

636 10. Scalar Diﬀraction Theory

amplitude of the diﬀracted ﬁeld is then

ψ =

sin

θ −

ka sin θ J

(ka sin θ) J

−

(ka sin θ) J

¶¸

. (10.52)

The dark rings are determined by

ka sin θJ

(ka sin θ)J

−

(ka sin θ)J

= 0. (10.53)

The series expansion of J

−

+ ···. (10.54)

For a ¿ w

, we only take the ﬁrst two terms of the expansion of J

and the

ﬁrst term of the expansion of J

, and (10.53) is expressed as

(x) −

− a

(x) = 0, (10.55)

where x = ka sin θ. The value of x for dark rings is determined by (10.55),

whereas the value of x for dark rings in diﬀraction of a uniform plane wave

is determined by J

(x) = 0 (x 6= 0). We can compare the diﬀerence between

them. Let x

be a root of J

(x) (x 6= 0), and x

+ ∆x be the solution of

(10.55), we have

[xJ

(x)]

x=x

∆x −

− a

(x)

x=x

∆x + J

)

= 0. (10.56)

Introducing the derivative of Bessel functions, we obtain

)∆x +

− a

)∆x − J

)] = 0. (10.57)

Because J

) = 0, (10.57) is simpliﬁed to

∆x =

− a

)

. (10.58)

Substitution of x = ka sin θ into (10.58) yields

∆θ =

− a

) sin 2θ

, (10.59)

10.3 Diﬀraction of Gaussian Beams 637

0 2 4 6 8

0.0

1.0

Plane wave

Gauss beam

k a sin

1 wa

Figure 10.10: Comparison of the intensity distributions of Fraunhofer diﬀrac-

tion of Gaussian beam and plane wave.

where θ

is the diﬀraction angle corresponding to a dark ring for plane-wave

incidence. Expression (10.59) shows that the diﬀraction angle of a Gaussian

beam is larger than that of a plane wave, and the lower the diﬀraction order,

the larger the diﬀerence between the diﬀraction angles is. From (10.59) we

can derive the relative diﬀerence of the diﬀraction angles between a Gaus-

sian beam and a plane wave. As the diﬀraction angle is small, we take the

approximation that sin θ = θ and cos θ = 1, and (10.59) is expressed as

∆θ

(kaθ

)

− 1

, (10.60)

where x

is the nth root of the Bessel function of the ﬁrst order. In Table

10.1, we give the ﬁrst three roots of J

(x) and the relatively increasing quan-

tities of the diﬀraction angles of a Gaussian beam with respect to those of a

plane wave for diﬀerent a/w

In Fig. 10.10, the solid curve represents the intensity of Fraunhofer diﬀrac-

tion of a Gaussian beam at a circular aperture, and the dashed curve repre-

sents that of a plane wave at the same aperture.

Table 10.1 x

and ∆θ/θ

∆θ/θ

a/w

= 0.5 a/w

= 0.25

= 3.832 4.5% 0.9%

= 7.016 1.35% 0.27%

= 10.173 0.64% 0.13%

638 10. Scalar Diﬀraction Theory

Figure 10.11: Fresnel diﬀraction of a Gaussian beam at a circular aperture.

10.3.2 Fresnel Diﬀraction of Gaussian Beams

As in the discussion of the Fresnel diﬀraction of a plane wave, here we discuss

only the diﬀraction intensity on the axis. In Fresnel diﬀraction there is no

requirement that the aperture must be located at the beam waist, so in the

aperture the amplitude and phase of the incident wave are both variable.

It is easy to prove that three diﬀraction formulas will give the same results

under the paraxial condition, and here we choose the Rayleigh–Sommerfeld

diﬀraction integral of the ﬁrst kind. For normal incidence the inclination

factor in the integral is omitted.

In Fig. 10.11, a Gaussian beam whose beam waist is located at L

left

of the ap erture is normally incident on it. The radius of the beam waist is

, and the radius of the aperture is a. From (10.17) the Fresnel diﬀraction

integral at the observation point P on the axis is

ψ(z) =

exp

−jkz

1 −

¶¸

dρ

, (10.61)

where z is the distance from P to the aperture center. The amplitude of the

incident beam in the aperture is

+ js

exp

−jk

2(L

+ js)

¸¾

, (10.62)

where A is a constant and s is the confocal parameter of the incident beam.

Substituting (10.62) into (10.61), we obtain

ψ =

+z+js

exp[−jk(L

+z)]

1−exp

−

jka

+js

¶¸¾

. (10.63)

(10.63) can also be expressed as

ψ = ψ

1 − exp

−

jka

+ js

¶¸¾

, (10.64)

10.3 Diﬀraction of Gaussian Beams 639

where ψ

is the amplitude at the observation point in the absence of the

opaque screen. (10.64) can be further expressed as

ψ = ψ

1 − exp

−

jka

¶¸¾

, (10.65)

where

= w

1 +

, R

+ s

. (10.66)

In the above formulas, w

is the radius of the beam waist, and w

and R

are

the beam radius and the curvature radius at the aperture, respectively. The

intensity is then

I = 2I

exp

−a

¶½

cosh

− cos

¶¸¾

, (10.67)

where I

is the intensity at P in the absence of the screen. As a/w

¿ 1, the

diﬀraction intensity is simpliﬁed to

I = 2I

1 − cos

¶¸¾

= 4I

sin

¶¸

. (10.68)

Obviously the intensity is dependent on the radius of the aperture. Varying

the radius causes entire extinction at the observation point, and the condition

for this is

= nπ, (10.69)

where n is an integer. Under the condition that

n +

π, (10.70)

the diﬀraction intensity is maximum. If the condition that a/w

¿ 1 is not

satisﬁed, (10.67) can be expressed as

I = 2I

exp

−a

(

1 +

− cos

¶¸

)

. (10.71)

Under the condition of (10.69), the intensity is minimum, but it is not en-

tirely extinct. As the radius of the aperture increases, the maximum and

the minimum of the diﬀraction intensity are both close to I

. Figure 10.12

shows the diﬀraction intensity at a point on the beam axis as a function of

the radius of the aperture.

640 10. Scalar Diﬀraction Theory

0.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

1.0

1.2

1.4

1.6

'wa

Figure 10.12: The diﬀraction intensity at a point on the beam axis as a

function of the radius of the aperture.

10.4 Diﬀraction of Plane Waves

in Anisotropic Media

In microwaves and optoelectronics, more and more crystal materials are used

to give rise to some particular functions which involve generation of mi-

crowave and light wave, propagation, transformation, and interaction with

other kinds of waves, etc. Most crystals are anisotropic, so it is valuable to

study the diﬀraction in anisotropic media.

In this section we will investigate the diﬀraction in uniaxial crystals. Since

the diﬀraction of ordinary wave is the same as that in isotropic media, we

discuss only the diﬀraction of the extraordinary wave.

10.4.1 Fraunhofer Diﬀraction at Square Apertures

In Fig. 10.13, a monochromatic plane wave with angular frequency ω is in-

cident normally on a square aperture of side 2a on the surface of a uniaxial

crystal from free space. The coordinate system is so chosen that its z axis

coincides with the optical axis of the crystal, the x axis and y axis are along

other principal dielectric axes. The angle between the incident direction and

the optical axis is α. In order to process more conveniently, the y axis is set

to be along a side of the aperture.

In crystals, if the coordinate axes are along the principal dielectric axes,

this coordinate system is called the principal coordinate system. The wave

equations for an extraordinary wave with angular frequency ω in the principal

coordinator system is given in (9.215) as:

∂

∂x

∂

∂y

∂

∂z

+ k

ψ = 0, (10.72)