Chapter Questions
Determine the Fourier transform of the function$$E(x)=\left\{\begin{array}{ll}E_{0} \sin k_{p} x & |x| < L \\0 & |x|>L\end{array}\right.$$Make a sketch of $\mathscr{F}\{E(x)\} .$ Discuss its relationship to Fig. 11.11
Determine the Fourier transform of$$f(x)=\left\{\begin{array}{ll}\sin ^{2} k_{p} x & |x| < L \\0 & |x| > L\end{array}\right.$$Make a sketch of it.
Determine the Fourier transform of$$f(t)=\left\{\begin{array}{ll}\cos ^{2} \omega_{p} t & |t| < T \\0 & |t| > T\end{array}\right.$$Make a sketch of $F(\omega),$ then sketch its limiting form as $T \rightarrow \pm \infty$
Show that $\mathscr{F}\{1\}=2 \pi \delta(k)$
Determine the Fourier transform of the function $f(x)=$ $A \cos k_{0} x$
Consider the function$$E(t)=E_{0} e^{-i \omega_{0} t} e^{-t^{2} / 2 \tau^{2}} $$ and first check that the exponents are unitless. Then show that the Fourier transform of $E(t)$ is$$E(\omega)=\sqrt{2 \pi} E_{0} \tau e^{-\tau^{2}\left(\omega-\omega_{0}\right)^{2} / 2}$$
$$\int_{-\infty}^{+\infty} e^{-a x^{2}+b x+c} d x=\left(\frac{\pi}{a}\right)^{1 / 2} e^{\frac{1}{4}\left(b^{2} / a\right)+c}$$
With the previous problem in mind show that the inverse transform of$$E(\omega)=\sqrt{2 \pi} E_{0} \tau e^{-\tau^{2}\left(\omega-\omega_{0}\right)^{2} / 2}$$brings you back to $E(t)$
Show that if $f(x)$ is real and even, its transform is real and even. [Hint: Start with Eq. (11.5), use the Euler formula from Section 2.5, and assume that $f(x)$ has both a real and an imaginary part.
Given that $\mathscr{F}\{f(x)\}=F(k)$ and $\mathscr{F}\{h(x)\}=H(k),$ if $a$ and $b$ are constants, determine $\mathscr{F}\{a f(x)+b h(x)\}$
Figure $P .11 .10$ shows two periodic functions, $f(x)$ and $h(x)$ which are to be added to produce $g(x)$. Sketch $g(x)$; then draw diagrams of the real and imaginary frequency spectra, as well as the amplitude spectra for each of the three functions.
Compute the Fourier transform of the triangular pulse shown in Fig. P.1 1.11 . Make a sketch of your answer, labeling all the pertinent values on the curve.
Given that $\mathscr{F}\{f(x)\}=F(k),$ introduce a constant scaling factor $1 / a$ and determine the Fourier transform of $f(x / a) .$ Show that the transform of $f(-x)$ is $F(-k)$
Show that the Fourier transform of the transform, $\mathscr{F}\{f(x)\}$ equals $2 \pi f(-x)$, and that this is not the inverse transform of the transform, which equals $f(x)$. This problem was suggested by Mr. D. Chapman while a student at the University of Ottawa.
The rectangular function is often defined as$$\operatorname{rect}\left|\frac{x-x_{0}}{a}\right|=\left\{\begin{array}{ll}0, & \left|\left(x-x_{0}\right) / a\right| > \frac{1}{2} \\\frac{1}{2}, & \left|\left(x-x_{0}\right) / a\right|=\frac{1}{2} \\1, & \left|\left(x-x_{0}\right) / a\right| < \frac{1}{2}\end{array}\right.$$where it is set equal to $\frac{1}{2}$ at the discontinuities (Fig. P.11.14). Determine the Fourier transform of$$f(x)=\operatorname{rect}\left|\frac{x-x_{0}}{a}\right|$$Notice that this is just a rectangular pulse, like that in Fig. $11.1 b$, shifted a distance $x_{0}$ from the origin.
With the last two problems in mind, show that $\mathscr{F}\{(1 / 2 \pi) \times$ $\left.\operatorname{sinc}\left(\frac{1}{2} x\right)\right\}=\operatorname{rect}(k),$ starting with the knowledge that $\mathscr{F}\{\operatorname{rect}(x)\}=$$\operatorname{sinc}\left(\frac{1}{2} k\right),$ in other words, Eq. (7.58) with $L=a,$ where $a=1$
Utilizing Eq. (11.38), show that $\mathscr{F}^{-1}\{\mathscr{F}\{f(x)\}\}=f(x)$
Given $\mathscr{F}\{f(x)\},$ show that $\mathscr{F}\left\{f\left(x-x_{0}\right)\right\}$ differs from it only by a linear phase factor.
Prove that $f \otimes h=h \oplus f$ directly. Now do it using the convolution theorem.
Prove that the area under the convolution of the functions $f(x)$ and $h(x)$ equals the product of the areas under each of those functions.
Examine the three graphs in Fig. P. 11.20 and explain what's being illustrated. Discuss how the shape of $g(X)$ arises. Why is $g(X)$ symmetrical about $X=0 ?$ What's the significance of the width of $g(x) ?$ Compute the peak value of $g(x)$
Suppose we have two functions, $f(x, y)$ and $h(x, y),$ where both have a value of 1 over a square region in the $x y$ -plane and are zero everywhere else (Fig. P.11.21). If $g(X, Y)$ is their convolution, make a plot of $g(X, 0)$
Referring to the previous problem, justify the fact that the convolution is zero for $|X| \geq d+\ell$ when $h$ is viewed as a spread function.
Use the method illustrated in Fig. 11.30 to convolve the two functions depicted in Fig. P.1 1.23
Given that $f(x) \oplus h(x)=g(X)$, show that after shifting one of the functions an amount $x_{0},$ we get $f\left(x-x_{0}\right) \oplus h(x)=g\left(X-x_{0}\right)$
Figure $P .11 .25$ depicts a single "saw tooth" function and its convolution. Note that the convolution is asymmetrical-explain why that's reasonable. Why does the convolution begin at $0 ?$ How wide is the convolution and how does that relate to $f(x) ?$
Graphically convolve the two functions $f(x)$ and $h(x)$ shown in Fig. $P .11 .26$ How wide will the convolution be? Will it be symmetrical? Where will it start?
Prove analytically that the convolution of any function $f(x)$ with a delta function, $\delta(x)$, generates the original function $f(X)$
Prove that $\delta\left(x-x_{0}\right) \oplus f(x)=f\left(X-x_{0}\right)$ and discuss the meaning of this result. Make a sketch of two appropriate functions and convolve them. Be sure to use an asymmetrical $f(x)$
Show that $\mathscr{F}\left\{f(x) \cos k_{0} x\right\}=\left[F\left(k-k_{0}\right)+F\left(k+k_{0}\right)\right] / 2$ andthat $\mathscr{F}\left\{f(x) \sin k_{0} x\right\}=\left[F\left(k-k_{0}\right)-F\left(k+k_{0}\right)\right] / 2 i$
Figure $P .11 .30$ shows two functions. Convolve them graphically and draw a plot of the result
Graphically convolve, at least approximately, the two functions shown in Fig. P.11.31. Does that solution remind you of anything? Why is the convolution symmetrical? When does its peak value occur in relation to $f(x)$ and $h(x) ?$ How wide is the convolution? Why?
Given the function $$f(x)=\operatorname{rect}\left|\frac{x-a}{a}\right|+\operatorname{rect}\left|\frac{x+a}{a}\right|$$determine its Fourier transform. (See Problem 11.14 .)
Given the function $f(x)=\delta(x+3)+\delta(x-2)+\delta(x-5)$ convolve it with the arbitrary function $h(x)$
Make a sketch of the function arising from the convolution of the two functions depicted in Fig. P.11.34.
Figure $P .11 .35$ depicts a rect function (as defined above) and a periodic $c o m b$ function. Convolve the two to get $g(x)$. Now sketch the transform of each of these functions against spatial frequency $k / 2 \pi=1 / \lambda .$ Check your results with the convolution theorem. Label all the relevant points on the horizontal axes in terms of $d-$ like the zeros of the transform of $f(x)$
Figure $P .11 .36$ shows, in one dimension, the electric field across an illuminated aperture consisting of several opaque bars forming a grating. Considering it to be created by taking the product of a periodic rectangular wave $h(x)$ and a unit rectangular function $f(x)$ sketch the resulting electric field in the Fraunhofer region.
Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a center of symmetry.
Suppose a given aperture produces a Fraunhofer field pattern $E(Y, Z)$. Show that if the aperture's dimensions are altered such that the aperture function goes from $\mathscr{A}(y, z)$ to $\mathscr{A}(\alpha y, \beta z),$ the newly diffracted field will be given by $$E^{\prime}(Y, Z)=\frac{1}{\alpha \beta} E\left(\frac{Y}{\alpha}, \frac{Z}{\beta}\right)$$
Show that when $f(t)=A \sin (\omega t+\varepsilon), C_{f f}(\tau)=\left(A^{2} / 2\right) \cos \omega t$which confirms the loss of phase information in the autocorrelation.
Suppose we have a single slit along the $y$ -direction of width $b$ where the aperture function is constant across it at a value of $\mathscr{A}_{0} .$ What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from $\mathscr{A}_{0}$ at the center to 0 at $\pm b / 2$ via a cosinusoidal dropoff
How wide will it be? At what value of $x$ will the correlation peak? What is the maximum value of $c_{f h}(x) ?$ Is it symmetrical? [Hint: Slide either one over the other. Graphically find the cross-correlation $c_{f h}(x)$ of the two functions shown here:
Consider the periodic function$$f(x)=\cos (k x+\epsilon)$$ where the amplitude is $1.0,$ and $\epsilon$ is an arbitrary phase term. Show that the autocorrelation function (before being normalized) is$$c_{f f}(x)=\frac{1}{2} \cos k x$$
A rectangular pulse extends from $-x_{0}$ to $+x_{0}$ and has a height of $1.0 .$ Sketch its autocorrelation, $c_{f f}(X) .$ How wide is $c_{f f}(X) ?$ Is it an even or odd function? Where does it start (become nonzero) and where does it end?
Figure $P .11 .44$ depicts a single "saw tooth" function and its autocorrelation. Explain why $c_{f f}(X)$ is symmetrical about the origin. Why does it extend from -1 to $+1 ?$ Draw sketches where appropriate.
Show, from the integral definitions, that $f(x) \star g(x)=$ $f(x) \oplus g(-x),$ where the functions are real
Figure $P .11 .46$ depicts a function $f(x) .$ Draw, to scale, its autocorrelation function $c_{f f}(X) .$ How wide is $c_{f f}(X) ?$ How wide is each individual peak composing $c_{f f}(X) ?$
Figure $P .11 .47$ shows a function $f(x)$ consisting of a periodic array of equally spaced delta functions. Construct its autocorrelation and discuss whether or not it is periodic.
Imagine two uniformly illuminated small circular holes in an opaque screen, as shown in Fig. P.11.48. Construct its autocorrelation. Discuss the irradiance distribution for each resulting individual patch of light in the autocorrelation. Indicate the relative irradiances of the several patches of light in the autocorrelation. Discuss the overall size of the autocorrelation compared to the original function.
$^{*}$ Figure $P .11 .49$ shows a transparent ring on an otherwise opaque mask. Make a rough sketch of its autocorrelation function, taking $l$ to be the center-to-center separation against which you plot that function.
Consider the function in Fig. 11.49 as a cosine carrier multiplied by an exponential envelope. Use the frequency convolution theorem to evaluate its Fourier transform.