The Dirichlet Kernel

Consider a screen with equally spaced slits illuminated by a laser beam. Assuming the width of the slits is small compared to their separation, each one of them acts as a source of radiation as depicted in Figures 2.1 and 2.2.

Figure 2.1: Dirichelet kernels of order $N=0$ and $N=1$ .

Figure 2.2: Dirichelet kernels of order $N=2$ and $N=3$ . Each of these kernels is the observed far field amplitude profile of the radiation emitted coherently from an odd ($2N+1$ ) number of sources having equal strength. The observed intensity, which is perceived by the human eye, is proportional to the square of the amplitude. The separation between the source screen and the observation screen is much larger than is implied by the picture.

Observations show that at large distances from these sources - large compared to the laser wave length, and large compared to the slit separation - the emitted radiation forms a Fraunhofer diffraction which is characterized by the number of slit sources. For an odd number, say $2N+1=1,3,5,7$ of them, the measured amplitude profiles and the observed intensity (= squared amplitude) profiles have forms which are shown in Figures 2.1-2.2

Each of the diffraction amplitude profiles is the interference pattern of radiation from an odd number of slit sources having equal amplitude and equal phase. Each pattern can be represented with mathematical precision as a finite Fourier sum with a number terms equal to the odd number of slit sources causing the pattern.

If the number of sources is $2N+1$ , then the corresponding pattern is called a Dirichelet kernel of integral order $N$ . Its mathematical form is

$$\delta_N(u)=\frac{1}{2\pi}\frac{\sin (N+\frac{1}{2})u}{\sin \frac{u}{2}}~,$$ (24)

where $u$ is the displacement along the screen where the pattern is observed. This kernel is a fundamental concept. Among others, it is the mathematical root of the Fourier Series theorem and the sampling theorem, applications which we shall develop as soon as we have defined this kernel mathematically.

Remark 1. Q: What can one say about diffraction patterns caused by an even number of slit sources?
A: The essential difference from an odd number lies in the observed amplitude. Whereas for an odd-numbered source the peak amplitude has always the same sign every period, for an even-numbered slit source the peak amplitude alternates between positive and negative from one period to the next. See Figure 2.3. However, such an amplitude is still given by Eq.(2.4), provided $N$ assumes odd integer values, $\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots$ . Such an amplitude pattern is called a Dirichelet kernel of odd half-integral order.

Figure 2.3: Dirichelet kernels of odd half-integral order $N=\frac {1}{2},\frac {3}{2},\frac {5}{2}$ . These kernels differ from those in Figures 2.1-2.2 in that here the number of sources is even ($2N+1$ ). Furthermore, the observed amplitude has peaks that alternate in sign every period.

Remark 2. Q: What happens to the diffraction pattern if each slit has a finite width, say $w$ ?
A: In that case the diffraction pattern gets modulated (i.e. multiplied) by the sinc function

$$\frac{\sin (u/w)}{u/w}~.$$

This conclusion is validated by means of the Fourier Integral Theorem, which is developed in the next Section 2.3.1 on page .

The Dirichlet kernel of integral order arises in the context of Fourier series whose orthonormal basis functions on $[0,2\pi ]$ are

$$\{ u_k (x)\} = \left\{ \frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}\cos nx, \frac{1}{\sqrt{\pi}}\sin nx\right\}\,.$$

Consider the $N^{\textrm{th}}$ partial sum $S_N$ of $f$ , a function integrable on the interval $[0,2\pi ]$

$$S_N = \frac{a_0}{2}+\sum^N_{n=1} a_n\cos nx +\sum^N_{n=1} b_n \sin nx$$

$$S_N(x) = \frac{1}{\pi}\int^{2\pi}_0 \left[ \frac{1}{2} +\sum^N_{n=1}\cos nx\cos nt+\sum^N_{n=1}\sin nx\sin nt\right] f(t)dt$$

$$= \int^{2\pi}_0 \frac{1}{\pi} \left[\frac{1}{2}+\sum^N_{n=1}\cos n(x-t) \right] f(t)dt$$

$$\equiv \int^{2\pi}_0 \delta_N(x-t) f(t)dt\,.$$

This is the (optimal) least squares approximation of $f$ .

Definition: (Dirichlet kernel $=$ ``periodic finite impulse function'') The function

$$\delta_N(u)=\frac{1}{\pi}\left[\frac{1}{2}+\sum^N_{n=1}\cos nu\right] = \frac{1}{2\pi}\sum^N_{n=-N} e^{inu}~~\qquad~~\hbox{with}~~u=x-t$$ (25)

is called the Dirichlet kernel and it is also given by

$$\frac{1}{2\pi}\,\frac{e^{-iNu}-e^{i(N+1)u}}{1-e^{iu}} = \frac{1}{... ... \,\frac{\sin \left(N+\frac{1}{2}\right) u}{\sin \frac{u}{2}} = \delta_N (u)\,.$$ (26)

Lecture 11