Diffraction from Circular Aperture

Consider the diffraction pattern of a circular aperture of radius $a$ whose center lies at $x=y=0$. (See Section 10.10.) We expect the pattern to be rotationally symmetric about the $z$-axis. In other words, we expect the intensity of the illumination on the projection screen to be only a function of the radial coordinate $r'=(x'^{\,2}+y'^{\,2})^{1/2}$. It is helpful to redefine the dimensionless parameters $u$ and $v$ as follows:

$$u = \frac{2\pi}{\lambda}\,\frac{a}{R}\,a,$$ (10.139)

$$v = \frac{2\pi}{\lambda}\,\frac{a}{R}\,r'.$$ (10.140)

Thus, $u$ now parameterizes the aperture radius, whereas $v$ is a normalized radial coordinate on the projection screen. Note, from Equation (10.96), that the far-field limit corresponds to $u\lesssim 1$, whereas the near-field limit corresponds to $u\gg 1$. Furthermore, a point on the projection screen lies in the geometric (i.e., as predicted by geometric optics) lit part of the screen if $v<u$, and vice versa. Finally, the aperture function takes the form

$$f(v) = \left\{ \begin{array}{lll} 1&\mbox{\hspace{0.5cm}}& \mbox{$v<u$\ }\\ [0.5ex] 0&&\mbox{otherwise}\end{array}\right..$$ (10.141)

When expressed in terms of the new variables, Equations (10.103) and (10.104) transform to give

$$f_c(u,v) = u\int_0^1\oint \cos\left(\frac{v^{\,2}}{2\,u} + \frac{u\,z^{\,2}}{2}-v\,z\,\cos\theta\right)z\,dz\,\frac{d\theta}{2\pi},$$ (10.142)

$$f_s(u,v) = u\int_0^1\oint \sin\left(\frac{v^{\,2}}{2\,u} + \frac{u\,z^{\,2}}{2}-v\,z\,\cos\theta\right)z\,dz\,\frac{d\theta}{2\pi},$$ (10.143)

where $z=(x^{\,2}+y^{\,2})^{1/2}/a$. Now, (Abramowitz and Stegun 1965)

$$\oint \cos(v\,z\,\cos\theta)\,\frac{d\theta}{2\pi} = J_0(v\,z),$$ (10.144)

$$\oint \sin(v\,z\,\cos\theta)\,\frac{d\theta}{2\pi} =0,$$ (10.145)

where (ibid.)

$$J_n(z)=\frac{1}{\pi}\int_0^\pi \cos(z\,\sin\theta-n\,\theta)\,d\theta$$ (10.146)

denotes a Bessel function of degree $n$. Hence, making use of some trigonometric identities (see Appendix B), Equations (10.142) and (10.143) reduce to

$$f_c(u,v) = \cos\left(\frac{v^{\,2}}{2\,u}\right)\,{\cal C}(u,v) -\sin\left(\frac{v^{\,2}}{2\,v}\right){\cal S}(u,v),$$ (10.147)

$$f_s(u,v) = \sin\left(\frac{v^{\,2}}{2\,u}\right)\,{\cal C}(u,v) + \cos\left(\frac{v^{\,2}}{2\,v}\right){\cal S}(u,v),$$ (10.148)

where

$${\cal C}(u,v) = u\int_0^1 \cos\left(\frac{u\,z^{\,2}}{2}\right)J_0(v\,z)\,z\,dz,$$ (10.149)

$${\cal S}(u,v) = u\int_0^1 \sin\left(\frac{u\,z^{\,2}}{2}\right)J_0(v\,z)\,z\,dz.$$ (10.150)

It is helpful, at this stage, to introduce the so-called Lommel functions (of two arguments) (Watson 1962)

$$U_n(u,v) =\sum_{s=0,\infty} (-1)^s\left(\frac{u}{v}\right)^{n+2s}J_{n+2s}(v),$$ (10.151)

$$V_n(u,v) =\sum_{s=0,\infty} (-1)^s\left(\frac{v}{u}\right)^{n+2s}J_{n+2s}(v).$$ (10.152)

In the geometric lit region, $v<u$, the integrals ${\cal C}(u,v)$ and ${\cal S}(u,v)$ are conveniently expanded in terms of the convergent $V_n$ Lommel functions (Born and Wolf 1980)

$${\cal C}(u,v) =\sin\left(\frac{v^{\,2}}{2\,u}\right)+\sin\left(\frac{u}{2}\right)V_0(u,v)-\cos\left(\frac{u}{2}\right)V_1(u,v),$$ (10.153)

$${\cal S}(u,v) =\cos\left(\frac{v^{\,2}}{2\,u}\right)-\cos\left(\frac{u}{2}\right)V_0(u,v)-\sin\left(\frac{u}{2}\right)V_1(u,v).$$ (10.154)

(See Exercise 22.) Likewise, in the geometric shadow region, $v>u$, the integrals can be expended in term of the convergent $U_n$ Lommel functions (Born and Wolf 1980)

$${\cal C}(u,v) =\cos\left(\frac{u}{2}\right)U_1(u,v)+\sin\left(\frac{u}{2}\right)U_2(u,v),$$ (10.155)

$${\cal S}(u,v) =\sin\left(\frac{u}{2}\right)U_1(u,v)-\cos\left(\frac{u}{2}\right)U_2(u,v).$$ (10.156)

(See Exercise 22.) It follows (with the aid of some trigonometric identities) that

$$f_c(u,v) = \sin\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)V_0(u,v)- \cos\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)V_1(u,v),$$ (10.157)

$$f_s(u,v) =1- \cos\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)V_0(u,v)- \sin\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)V_1(u,v)$$ (10.158)

when $v<u$, and

$$f_c(u,v) = \cos\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)U_1(u,v)+ \sin\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)U_2(u,v),$$ (10.159)

$$f_s(u,v) =\sin\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)U_1(u,v)- \cos\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)U_2(u,v)$$ (10.160)

when $v>u$.

Figure 10.20: Far/near-field diffraction pattern of a circular aperture. The left and right panels correspond to $u=1$ and $u=50$, respectively. The thick black line indicates the physical extent of the aperture.

Finally, making use of Equation (10.110), the previous four equations imply that the illumination intensity on the projection screen can be written

$$\frac{{\cal I}(v)}{{\cal I}_0}= \left[V_0(u,v)-\cos\left(\frac{u^... ...,2} + \left[V_1(u,v)-\sin\left(\frac{u^{\,2}+v^{\,2}}{2\,u}\right)\right]^{\,2}$$ (10.161)

when $v<u$, and

$$\frac{{\cal I}(v)}{{\cal I}_0}= \left[U_1^{\,2}(u,v)+U_2^{\,2}(u,v)\right]$$ (10.162)

when $v>u$. Here, ${\cal I}_0$ is the intensity of the light illuminating the aperture from behind.

Figure 10.20 shows a typical far-field (i.e., $u\lesssim 1$) and near-field (i.e., $u\gg 1$) diffraction pattern of a circular aperture, as determined from the previous analysis. It can be seen that the far-field diffraction pattern is similar in form to that predicted by the simplified Fourier analysis of Section 10.9. On the other hand, the near-field diffraction pattern is quite different. In fact, the near-field diffraction pattern is fairly similar in form to the geometric image of the aperture, apart from the presence of fringes within the image.