Appendix A FOURIER TRANSFORM This appendix gives the definition of the one-, two-, and three-dimensional Fourier transforms as well as their properties. A.l One-Dimensional Fourier Transform If we have a one-dimensional (1-D) function fix), its Fourier transform F(m) is de­ fined as F(m) = [ f(x)exp( -2mmx)dx, (A.l.l) where m is a variable in the Fourier space. Usually it is termed the frequency in the Fou­ rier domain. If xis a variable in the spatial domain, miscalled the spatial frequency. If x represents time, then m is the temporal frequency which denotes, for example, the colour of light in optics or the tone of sound in acoustics. In this book, we call Eq. (A.l.l) the inverse Fourier transform because a minus sign appears in the exponent. Eq. (A.l.l) can be symbolically expressed as F(m) =,r 1{Jtx)}. (A.l.2) where ~-~denotes the inverse Fourier transform in Eq. (A.l.l). Therefore, the direct Fourier transform in this case is given by f(x) = r~ F(m)exp(2mmx)dm, (A.1.3) which can be symbolically rewritten as fix)= ~{F(m)}. (A.l.4) Substituting Eq. (A.l.2) into Eq. (A.l.4) results in 200 Appendix A ftx) = 1"1"-'{.ftx)}. (A.l.5) Therefore we have the following unity relation: ,.,... = l. (A.l.6) It means that performing a direct Fourier transform and an inverse Fourier transform of a function.ftx) successively leads to no change. Using the identity exp(ix) = cosx + isinx and Eq. (A.l.l), one has F(m) = A(m)- iB(m), (A.l.7) where A(m) ~ J~ f(x)cos(2mnx)dx, { (A.l.8) B(m) = L f(x)sin(2nm.x)dx. A.2 Two-Dimensional Fourier Transform In a similar way, the direct and inverse Fourier transforms of a two-dimensional (2- D) function.ftx, y) can be expressed as f(x, y) = [ F(m,n)exp[21li(mx + ny)]dmdn, (A.2.l) and F(m,n) =[ f(x, y)exp[-2m(mx + ny)]dxdy, (A.2.2) respectively. If x andy are spatial coordinates, the exponent in Eq. (A.2.2), exp[-2m(mx + ny)], represents a plane wave if we recognize that k. =2nm and k, =21l'll. Here k, and k, are the components of the wave vector k in the x andy directions, i.e. (A.2.3) Appendix A 201 where .A is the illumination wavelength. In other words, the inverse Fourier transform of a spatial function is equivalent to resolving the function into a series of plane waves propagating in different directions and the direct Fourier transform means that the origi­ nal function is represented by a superposition of these plane waves each of which has a particular weighting factor. A.3 Three-Dimensional Fourier Transform For a three-dimensional (3-D) function fix, y, z), we have the direct and inverse 3-D Fourier transforms as follows: f(x, y, z) = [ F(m, n, s)exp[21ti(mx + ny + sz)]dmdnds, (A.3.1) and F(m, n, s) =[ f(x, y, z)exp[-2.ni(mx + ny +sz)]dxdydz. (A.3.2) y phase change between two .....---.....---consecutive wavefronts: 2x X z Fig. A.3.1 A plane wave corresponds to an exponent in Eq. (A.3.1). According to the discussion in Section A.2, the exponent in Eqs. (A.3.2) denotes a plane wave with the wave-vector components, k,, k, and k, which are given by 202 Appendix A k \ = 2JT:n, (A.3.3) k, = 2ns. If 27r(mx + ny + sz) =constant= A, this equation gives a series of parallel planes. The phase on these planes is constant. If A = 2~ (j = 0,±1,±2... ), the phase difference between two adjacent planes is 2tr. The spatial periods along the x, y and z axes are 2tr 1 A.=-=- x kx m, 2tr 1 A.,. =-k =-, (A.3.4) . " n 2tr 1 A.=-=- ' k, s Here Eq. (A.3.3) has been used. As we expect, m, n, and s correspond to the spatial fre­ quencies in the x, y and z directions, respectively. A spatial frequency vector m can be introduced with three components m, n, s. Thus the direct and inverse 3-D Fourier trans­ forms can be expressed in a compact form: f(x) = [ F(m)exp(211im •r)dm, (A.3.5) and F(m) =[f(r)exp(-2mm •r)dr, (A.3.6) where the vector r has three components x, y and z. A.4 Fourier Transform Theorems In this section, we give the Fourier transform theorems without giving the deriva­ tion procedure. The theorems are given for a 1-D case but the 2-D and 3-D forms of the theorems can be easily written down. Appendix A 203 a) Similarity theorem If F(m) =.1" {flx)}, then .1" {{(ax)}= _!_ F( m l_ (A.4.1) lal a) It means that a "stretching" of the coordinate in the x space leads to a contraction of the coordinates in the Fourier space and a change of the Fourier transform by a factor of lllal. b) Shift theorem If F(m) =.1"{flx)}, then !f {flx- a)} =F(m) exp( -21tima). (A.4.2) It means that a translation of a function in the x space leads to a linear phase shift in the Fourier space. c) Parsval's theorem If F(m) = .1"{f(x)}, then (A.4.3) which is a statement of the conservation of energy in physics. d) Convolution theorem If F(m) = !f{flx)} and G(m) = !f{g(x)), then !r{[Icc;)g(x-c;)dc;}= F(m)G(m). (A.4.4) or !F{J(x)® g(x)}= F(m)G(m). (A.4.5) This theorem implies that the Fourier transform of a convolution operation of two func­ tions in the x space is simply equivalent to the product of their corresponding Fourier transforms. 204 Appendix A e) Autocorrelation theorem If F(m) = ,1"{f(x)), then (A.4.6) or (A.4.7) It is noted that the autocorrelation theorem is a special case of the convolution theorem if we let g(x)= j'(-x). Appendix B HANKEL TRANSFORM In this appendix, a special form of the two-dimensional (2-D) Fourier transform, called the Hankel transform, is discussed. Let us start with the 2-D Fourier transform in Cartesian coordinates, which is given by F(m,n) =I r~f(x, y)exp[2m(xm+ yn)]dxdy. (B.l.l) The Hankel transform is the 2-D Fourier transform in a polar coordinate. The function f{x, y) can be represented by a functionf{r, ~) if the following coordinate transformation is adopted: x = rc~s(/), { (B.1.2) y = rsm(/), and m = l cos If/, { (B.1.3) n = l sin If/, where r and (/)are the polar coordinates in the x - y plane, whereas l and lfl are the polar coordinates in them- n plane. Therefore Eq. (B.l.l) can be rewritten as F(l,(J) = rr Jj(r,~)exp[21tir[ COS((/) -lf!)]rdrd(/). (B.l.4) 206 AppendixB In the case of a circularly symmetric system,Jtr, f/>) =f(r). Thus the Fourier transform of f(r) is also circularly symmetric, which may be denoted by F([). Finally, Eq. (B.l.4) can reduce to (B.l.S) which is called the Hankel transform. In this expression, I J 0 (x)=- J.2"exp(±ixcosq>)dq> (B.l.6) 2TC 0 is a Bessel function of the first kind of order zero. 0 2 4 6 8 10 X Fig. B.l.l Bessel functions of the first kind of the first five orders, J0(x), J1(x), h(x), h(x) and J4(x). Iff(r) is a uniform function within a radius a: f(r){ , r~ a, , r>a, (B.l.7) thus using the identity (B.l.8) Appendix B 207 one can derive an analytical expression for Eq. (B.I.5): (B.l.9) where l1(x) is a Bessel function of the first kind of order one. Fig. B.l.l gives the Bessel functions of the first kind of the first four orders. The function 2JI(x)lx is also termed the Airy function and is shown in Fig. B. l.2. 0.75 Airy function o.s y X Fig. B.l.2 Airy function 2J1(r)lr: 2-D behaviour. Appendix C DELTA FUNCTIONS This appendix summarizes the main properties of delta functions. For a one­ dimensional (1-D) problem, the definition of a delta function Ci(x) is given by oo X= 0, O(x) =j ' (C.l.l) 0 , X "#0, and [ 8(x)dx= 1. (C.l.2) Eqs. (C.l.l) and (C.l.2) give a complete definition of a delta function. Mathematically, a delta function represents an infinity at the origin (x =0). Physically, it represents an im­ pulse response or action. For example, in optical imaging, a point illumination source or a point detector can be represented by a delta function. In general, a delta function can be defined with respect to an arbitrary position x0• In this case, Eqs. (C. 1.1) and (C.l.2) should be rewritten as (C.l.3) and r~ 8(x-xo)c4 = 1. (C.l.4) One of the important properties of delta functions is mathematically expressed as 210 Appendix C [ D(x- x 0 )f(x)dx = f(x0 ), (C.l.5) or f~ 8(x)f(x)dx = /(0).