Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf
ВУЗ: Не указан
Категория: Не указан
Дисциплина: Не указана
Добавлен: 28.06.2024
Просмотров: 902
Скачиваний: 0
38 |
FUNDAMENTALS OF WAVE PROPAGATION |
where the real parts are actually the physical solutions. They can be more generally written as
Eðr; tÞ ¼ E0 cosðk r wtÞ
ð3:7-2Þ
Hðr; tÞ ¼ H0 cosðk r wtÞ;
where E0 and H0 have components (Ex,Ey), (Hx,Hy), and k, r in this case are simply
k ¼ 2p=l and z, respectively. |
|
|
@H |
|
|
|
Substituting Eqs. (3.7-1) in to r E ¼ m |
|
|
gives the following: |
|
||
@t |
|
|||||
kEy^ex kEx^ey ¼ mw½Hx^ex þ Hy^ey& |
ð3:7-3Þ |
|||||
Hence, |
|
|
|
|
|
|
Hx ¼ |
1 |
Ey |
|
|
||
|
|
|
||||
|
|
ð3:7-4Þ |
||||
|
|
|
|
|
|
|
1
Hy ¼ Ex;
where is called the characteristic impedance of the medium. It is given by
|
¼ k m ¼ |
|
m ¼ p |
ð |
|
Þ |
|
|
w |
v |
|
m=e |
|
3:7-5 |
|
|
|
|
|
||||
Note that |
|
|
|
|
|
|
|
E H ¼ ðExHx þ EyHyÞejðkzþwtÞ ¼ 0 |
ð3:7-6Þ |
||||||
Thus, the electric and magnetic fields are orthogonal to each other. |
|
|
|
||||
The Poynting vector S is defined by |
|
|
|
|
|
|
|
|
S ¼ E H; |
|
ð3:7-7Þ |
||||
which has units of W/m2, indicating power flow per unit area in the direction of propagation.
Polarization indicates how the electric field vector varies with time. Again assuming the direction of propagation to be z, the electric field vector including time dependence can be written as
E ¼ Real½ðExex þ EyeyÞejðkzþwtÞ& |
ð3:7-8Þ |
Ex and Ey can be chosen relative to each other as |
|
Ex ¼ Ex0 |
ð3:7-9Þ |
Ey ¼ Ey0 ej ; |
ð3:7-10Þ |
PLANE EM WAVES |
39 |
where Ex0 and Ey0 are positive scalars, and is the relative phase, which decides the direction of the electric field.
Linear polarization is obtained when ¼ 0 or p. Then, we get |
|
E ¼ ðEx0 ex Ey0 eyÞ cosðkz þ wtÞ |
ð3:7-11Þ |
In this case, ðEx0 ex Ey0 eyÞ can be considered as a vector that does not change in direction with time or propagation distance.
Circular polarization is obtained when ¼ p=2; and E0 ¼ Ex0 ¼ Ey0 . Then, we get
E ¼ E0 cosðkz þ wtÞex E0 sinðkz þ wtÞey |
ð3:7-12Þ |
We note that jEj ¼ E0. For ¼ p=2, E describes a circle rotating clockwise during propagation. For ¼ þp=2, E describes a circle rotating counterclockwise during propagation.
Elliptic polarization corresponds to an arbitrary . Equation (3.7-1) for the electric field can be written as
|
|
|
|
|
|
E ¼ Exex þ Eyey |
|
|
|
|
ð3:7-13Þ |
||||||||||||
where |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Ex ¼ Ex0 cosðkz þ wtÞ |
|
|
|
|
ð |
3 |
: |
7-14 |
Þ |
||||||||||
|
|
|
|
Ey ¼ Ey0 cosðkz þ wt Þ |
|
|
|
||||||||||||||||
Equation (3.7-14) can be written as |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
Ey |
¼ cosðkz þ wtÞ cos þ sinðkz þ wtÞ sin |
|
|
|
|
|
||||||||||||||||
|
|
|
|
|
|
|
|||||||||||||||||
|
Ey0 |
ð |
3:7-15 |
Þ |
|||||||||||||||||||
|
|
|
|
cos þ "1 |
|
|
# sin |
||||||||||||||||
|
|
|
E |
|
|
E |
2 |
||||||||||||||||
|
|
¼ |
x |
|
x |
|
|
|
|
|
|||||||||||||
|
|
Ex0 |
Ex0 |
|
|
|
|
|
|
||||||||||||||
This equation can be further written as |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
E02 |
|
2E0 |
E0 cos |
þ |
E02 |
¼ |
sin2 |
; |
ð |
3:7-16 |
Þ |
||||||||||
|
|
|
x |
|
x |
y |
y |
|
|
|
|
|
|
||||||||||
where |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Ex0 |
¼ |
Ex |
Ey0 ¼ |
Ey |
|
ð3:7-17Þ |
||||||||||||
|
|
|
|
|
|
; |
|
|
|
||||||||||||||
|
|
|
|
|
Ex0 |
Ey0 |
|
||||||||||||||||
Equation (3.7-17) is the equation of an ellipse.
40 |
FUNDAMENTALS OF WAVE PROPAGATION |
Suppose the field is linearly polarized and E is along the x-direction. Then, we can write
E ¼ E0ejðkzþwtÞex |
ð3:7-18Þ |
||||
|
E0 j kz |
|
wt |
||
H ¼ |
|
e ð |
þ |
|
Þey; |
|
|
||||
where Real [ ] is assumed from the context. Intensity or irradiance I is defined as the time-averaged power given by
|
|
2p=w |
|
E2 |
|
|
|
w |
ð0 |
|
|
||
I ¼ |
|
jSjdt ¼ em |
0 |
; |
ð3:7-19Þ |
|
2p |
2 |
|||||
where S is the Poynting vector. It is observed that the intensity is proportional to the square of the field magnitude. This will be assumed to be true in general unless otherwise specified.
4
Scalar Diffraction Theory
4.1INTRODUCTION
When the wavelength of a wave field is larger than the ‘‘aperture’’ sizes of the diffraction device used to control the wave, the scalar diffraction theory can be used. Even when this is not true, scalar diffraction theory has been found to be quite accurate [Mellin and Nordin, 2001]. Scalar diffraction theory involves the conversion of the wave equation, which is a partial differential equation, into an integral equation. It can be used to analyze most types of diffraction phenomena and imaging systems within its realm of validity. For example, Figure 4.1 shows the diffraction pattern from a double slit illuminated with a monochromatic plane wave. The resulting wave propagation can be quite accurately described with scalar diffraction theory.
In this chapter, scalar diffraction theory will be first derived for monochromatic waves with a single wavelength. Then, the results will be generalized to nonmonochromatic waves by using Fourier analysis and synthesis in the time direction.
This chapter consists of eight sections. In Section 4.2, the Helmholtz equation is derived. It characterizes the spatial variation of the wave field, by characterizing the time variation as a complex exponential factor. In Section 4.3, the solution of the Helmholtz equation in homogeneous media is obtained in terms of the angular spectrum of plane waves. This formulation also characterizes wave propagation in a homogeneous medium as a linear system. The FFT implementation of the angular spectrum of plane waves is discussed in Section 4.4.
Diffraction can also be treated by starting with the Helmholtz equation and converting it to an integral equation using Green’s theorem. The remaining sections cover this topic. In Section 4.5, the Kirchoff theory of diffraction results in one formulation of this approach. The Rayleigh–Sommerfeld theory of diffraction covered in Sections 4.6 and 4.7 is another formulation of the same approach. The Rayleigh–Sommerfeld theory of diffraction for nonmonochromatic waves is treated in Section 4.8.
Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy
Copyright # 2007 John Wiley & Sons, Inc.
41
42 |
SCALAR DIFFRACTION THEORY |
Figure 4.1. Diffraction from a double slit [Wikipedia].
4.2HELMHOLTZ EQUATION
Monochromatic waves have a single time frequency f. As
w |
¼ |
2pf |
ð4:2-1Þ |
||
k ¼ |
|
|
|
||
c |
c |
||||
the wavelength l is also fixed. For such waves, the plane wave solution discussed in Section 3.7 can be written as
uðr; tÞ ¼ AðrÞ cosðot þ ðrÞÞ |
ð4:2-2Þ |
where A(r) is the amplitude, and (r) is the phase at r. In phasor representation, this becomes
uðr; tÞ ¼ Re½UðrÞe jot& |
ð4:2-3Þ |
where the phasor U(r) also called the complex amplitude equals AðrÞej ðrÞ. Equation (4.2-3) is often written without explicitly writing ‘‘the real part’’ for the sake of simplicity as well as simplicity of computation as in the next equation.
Substituting u(r,t) into the wave equation (3.5-7) yields
ðr2 þ k2ÞUðrÞ ¼ 0 |
ð4:2-4Þ |
where k ¼ o=c. This is called the Helmholtz equation. It is same as Eq. (3.6-9). Thus, in the case of EM plane waves, UðrÞ represents a component of the electric field or magnetic field phasor, as discussed in Chapter 3. The Helmholtz equation is valid for all waves satisfying the nondispersive wave equation. For example, with acoustical and ultrasonic waves, UðrÞ is the pressure or velocity potential.
If Uðr; tÞ is not monochromatic, it can be represented in terms of its time Fourier transform as
1ð
Uðr; tÞ ¼ Uf ðr; f Þe j2pftdf |
ð4:2-5Þ |
1