Question

How does Fraunhoffer diffraction pattern due to a single slit differ from that of a circular
aperture?

Answer 1

Answer on Question#38829, Physics, Optics

How does Fraunhoffer diffraction pattern due to a single slit differ from that of a circular aperture?

Answer:

The Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object.

The diffraction at a single slit of width $d$ is shown in Figure 1. Diffraction occurs in all directions to the right of the slit.

Fig.1. Graph and image of single-slit diffraction

The pattern consists of a central bright fringe (band) flanked by much weaker maxima alternating with dark fringes.

The general condition for a minimum for a single slit is:

$$m \lambda = d \sin \theta$$

where $m = 1,2,3,4$ and so on

- $d$ is the width of the slit,

- $\theta$ is the angle of incidence at which the minimum intensity occurs, and

- $\lambda$ is the wavelength of the light

The intensity profile can be calculated using the Fraunhofer diffraction equation as

$$I (\theta) = I _ {0} \mathrm {s i n c} ^ {2} \left(\frac {d \pi}{\lambda} \sin \theta\right)$$

where

- $I(\theta)$ is the intensity at a given angle,

- $I_0$ is the original intensity, and

- the unnormalized sinc function above is given by $\operatorname{sinc}(x) = \sin(x) / (x)$ if $x \neq 0$ , and $\operatorname{sinc}(0) = 1$

The amplitude distribution for diffraction due to a circular aperture forms an intensity pattern with a bright central band surrounded by concentric circular bands of rapidly decreasing intensity (Airy pattern). The 1st maximum is roughly $1.75\%$ of the central intensity. $84\%$ of the light arrives within the central peak called the Airy disk.

Fig.2. An image of an Airy disk.

Far away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula:

$$\sin \theta \approx 1. 2 2 \frac {\lambda}{d}$$

or, for small angles, simply

$$\theta \approx 1. 2 2 \frac {\lambda}{d}$$

Where $\theta$ is in radians, $\lambda$ is the wavelength of the light and $d$ is the diameter of the aperture.

The variation in intensity with angle is given by

$$I (\theta) = I _ {0} \left(\frac {2 J _ {1} (k a \sin \theta)}{k a \sin \theta}\right) ^ {2}$$

where $a$ is the radius of the circular aperture, $k$ is equal to $2\pi/\lambda$ and $J_1$ is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

Answer. The diffraction due to a single slit forms a pattern with a bright central vertical band surrounded by vertical bands. The diffraction due to a circular aperture forms a pattern with a bright central circular band surrounded by concentric circular bands of rapidly decreasing intensity