OPTICAL DIFFRACTION
3.1 INTRODUCTION TO DIFFRACTION
CHAPTER 3
INTRODUCTION TO DIFFRACTION 39
FIGURE 3.1 Illustration of diffraction as per Huygens: (a) for a circular aperture and (b) two waves Interfering.
3.1.1 Description of Diffraction
Diffraction has been defined as behavior that cannot be explained by straight line rays [83]. Figure 3.1a shows such a case. If a coherent or single-frequency point source illuminates a card having a circular hole or aperture, a sharp shadow will not appear on a screen behind the card unless the screen is very close to the card. Instead, light passing through the aperture acts as a multitude of point sources, including those at the circular aperture edge. As a result, energy falls outside the shadow-casting region of the screen. The waves from these point sources interfere with each other as shown in Figure 3.1b, where pointsAandBmay be considered to be point sources on opposite edges of the aperture. The spherical curves represent the peaks of the emanating waves at one instant of time. Constructive interference occurs where the peaks from the two sources coincide. Destructive interference occurs where a peak of one corresponds to a negative peak or dip of the other. The interference effect known as diffraction shows up as a series of diffraction rings surrounding the directly illuminated region on the screen (Figure 3.1a). The rings get fainter with size. In 1818, Fresnel won a prestigious French prize for showing the wave nature of light, thought to settle the controversy of whether light was basically particles, as Newton had conjectured, or waves as evidenced by interference. Since quantum mechanics in the early 20th century, it is known that light can be viewed as either a particle or a wave. At lower frequencies wave properties dominate and at higher frequencies particle properties dominate. Light is at the transition so that for understanding interference we assume waves and for lasers we need to assume particles.
Figure 3.1a also shows the transverse light intensities across the center of the screen in thexdirection. At sufficient distance between aperture and screen, consistent with
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FIGURE 3.2 Fourier Transform of square pulse: (a) input time function, (b) Fourier transform of (a) and (c) magnitude Fourier transform of (a).
the Fraunhofer approximation [49] (Section 3.3.5), the intensity at the screen is the 2D spatial Fourier transform of the 2D function passing through the aperture, in this case, a Bessel function, the 2D Fourier transform for a circularly symmetric function.
If a piece of film with an image is placed on the aperture, the 2D Fourier transform of the image would be obtained at the screen. The distance required is several meters.
However, a converging source or the use of a lens permits the Fourier transform at much shorter distances [83].
3.1.2 Review of Fourier Transforms
First, we discuss temporal Fourier transforms in time, useful for pulses, and then the spatial Fourier transform used for diffraction. The one-dimensional Fourier transform transforms a function f(t) of time t in seconds into a function F(ω) of temporal frequencyωin radians per second:
output
F(w)=
t input
f(t) e−jwtdt (3.1)
Equation (3.1) may be viewed as correlating f(t) against different frequencies to determine how much of each frequencyF(ω) exists in the functionf(t). Temporal frequencyν=ω/(2π) is measured in cycles per second or hertz. If a pulse of height unity (Figure 3.2a) has durationT, the transform from equation (3.1) is (Figure 3.2b)
F(w)= T/2
−T/2
exp{−jωt}dt= exp{−jωt}
−jω
T/2
−T/2
= 2 sin(ωT/2)
ω (3.2)
The amplitude spectrum in equation (3.2) is zero or null whenωT/2=nπ, where nis an integer. The first null (the Rayleigh distance) occurs whenn=1, atωT/2=π orω=2π/T, or in frequency atf =ω/(2π)=1/T, as shown in Figure 3.2b. The magnitude of the Fourier transform of a square pulse or boxcar is shown in Figure 3.2c.
Intensity is magnitude squared.
INTRODUCTION TO DIFFRACTION 41
A 1D electromagnetic wave cos(ωt−kxx) oscillates in time with angular fre- quency ω radians per second and oscillates in 1D space x with spatial angular frequencykxradians per meter. In an analogous transverse water wave, a cork floating on a wave bobs up and down with frequencyωradians per second and a flash pho- tograph shows the spatial wave oscillating withkxradians per meter inxdirection.
So the 1D Fourier transform in the spatial domain may be written by analogy with equation (3.1) by replacing timetby space variablexand angular frequency in time ωradians per second by angular frequency in spacekxradians per meter:
F(kx)=
x
f(x)e−jkxxdx (3.3)
wherekxis the wave number in thexdirection in radians per meter andxis distance in meters. Spatial frequencyfx=2πkx has units of cycles per meter (or lines per meter). Physical space has more than one dimension; consequently, higher dimension transforms are of interest.
In considering beams or propagation of images between planes, we will consider 2D spatial Fourier transforms. A 1D spatial Fourier transform, equation (3.3), may be extended to 2D as follows:
2D output
F(kx, ky)=
x
y 2D input
f(x, y)
plane wave
e−j(kxx+kyy)dxdy (3.4) where spatial frequencieskxandky are in radians per meter inxandydirections, respectively. Note that the input wave pattern is decomposed into a set of plane waves with different propagation directions (kx, ky) and amplitudes. The 2D input pattern is correlated with each plane wave to giveF(kx, ky), the amount of each plane wave present in the input. A square shaped beam or source of width and heightSwill have a transform similar to that shown in Figure 3.2 inxandyin the 2D spatial frequency domain with null points at 1/S(SreplacesT).
There are many reasons for converting from the time or space domain into the temporal or spatial frequency domain. Sometimes data are more meaningful in the frequency domain. For example, in the temporal frequency domainω, we can filter out specific frequencies that we consider noise, such as a single-frequency jammer. In the space domain, multiple plane waves from different directions (from distant sources) will strike a sensor array from different directions. The spatial Fourier transform will separate the plane waves into multiple discrete points in the spatial frequency domain (Section 14.1). A single-direction jammer can be removed in adaptive beamforming to prevent it from masking target information from a different direction. If a 2D transform is used with timeton one axis and spacex(or angle) on the other, sources with different frequencies and directions will show up as different points on the 2D Fourier transform.
In image processing, we can remove spatially periodic background noise;
for example, bars in front of a tiger in a cage can be filtered out in the spatial
42 OPTICAL DIFFRACTION
frequency domain. On taking the inverse transform, the tiger can be seen without the bars. Targets with spatial periodicity can be recognized, and early submarines could be identified by the periodic ribs in their structure.
Correlation and convolution are widely performed computations. The Fourier transform is normally used to perform convolution or correlation on a computer or in optics because convolution and correlation become multiplication in the frequency domain and the low cost of the fast Fourier transform in computers or the optical Fourier transform in optics makes the computation faster than a straight convolution or correlation computation [83]. Correlation is the basis for comparing images or searching for patterns or words in huge databases. Convolution implements linear systems such as smoothing, prediction, and many other signal processing functions.
Chapter 14 of my previous book [83] describes an optical data flow computer that I designed as a PI to DARPA and that was later partially constructed for NSA. It performs ultrafast correlation.