1.4 Basic Physical Optics (E-IV-2)

B. Fraunhofer and Fresnel diffraction

In general, if the observation screen is far removed from the slit on which plane waves fall (as in Figure 4-15) or a lens is used to focus the collimated light passing through the slit onto the screen (as in Figure 4-14), the diffraction occurring is described as Fraunhofer diffraction, after Joseph von Fraunhofer (1787-1826), who first investigated and explained this type of so-called far-field diffraction. If however, no lens is used and the observation screen is near to the slit, for either incident plane or spherical waves, the diffraction is called Fresnel diffraction, after Augustin Fresnel (1788-1829), who explained this type of near-field diffraction. The mathematical calculations required to determine the details of a diffraction pattern and account for the variations in intensity on the pattern are considerably more complicated for Fresnel diffraction than for Fraunhofer diffraction, so typically one studies first the Fraunhofer diffraction patterns, as we have.

Without going into the details of how to distinguish mathematically between Fresnel and Fraunhofer diffraction we can give results that help you decide whether the diffraction pattern formed is Fraunhofer or Fresnel in origin. Knowing this distinction helps you choose which equations to use in describing a particular diffraction pattern arising from a particular optical setup.

1. Criteria for far-field and near-field diffraction. Figure 4-16 shows the essential features of a general diffraction geometry, involving a source of light of wavelength l, an opening to "obstruct" the light, and a screen to form the diffraction pattern.

Figure 4-16 General diffraction geometry involving source, aperture, and screen

The distance from source to aperture is denoted as Z and that from aperture to screen as Z¢. Calculations based on geometries that give rise to Fraunhofer and Fresnel diffraction patterns verify the following:

If the distance Z from source to aperture and the distance Z¢ from aperture to screen are both greater than the ratio by a factor of 100 or so, the diffraction pattern on the screen is characteristic of Fraunhofer diffraction—and the screen is said to be in the far field. For this situation, all Fraunhofer-derived equations apply to the details of the diffraction pattern.
If either distance—Z or Z¢—is of the order of, or less than, the ratio , the diffraction pattern on the screen is characteristic of Fresnel diffraction and is said to be in the near field. For this situation, all Fresnel-derived equations apply to the details of the diffraction pattern.
Equation 4-22 indicates the "rule-of-thumb" conditions to be satisfied for both Z and Z¢ for Fraunhofer diffraction.

(4-22)

Figure 4-18 illustrates these conditions and shows the locations of the near field, far field, and a gray area in between. If the screen is in the gray area and accuracy is important, a Fresnel analysis is usually applied. If the screen is in the gray area and approximate results are acceptable, a Fraunhofer analysis (significantly simpler than a Fresnel analysis) can be applied.

Figure 4-17 Defining near-field and far-field regions for diffraction

Figure 4-18 shows how we can satisfy the conditions for Fraunhofer diffraction, as spelled out in Equation 4-22, through the use of focusing lenses on both sides of the aperture (Figure 4-18a)—or with a laser illuminating the aperture and a focusing lens located on the screen side of the aperture (Figure 4-18b). Either optical arrangement has plane waves approaching and leaving the aperture, guaranteeing that the diffraction patterns formed are truly Fraunhofer in nature.

Figure 4-18 Optical arrangements for Fraunhofer diffraction

Now let's see how Equation 4-22 and Figure 4-18 are applied in a real situation.

Example 8

Minati, a photonics technician, has been asked to produce a Fraunhofer diffraction pattern formed when light from a HeNe laser (l = 633 nm) passes through a pinhole of 150-mm diameter. In order to set up the correct geometry for Fraunhofer diffraction, Minati needs to know (a) the distance Z from the laser to the pinhole and (b) the distance Z¢ from the pinhole to the screen.

Solution: Minati needs first to test the conditions given in Equation 4-22 so she calculates the ratio of assuming the pinhole to be circular.

Ratio = 0.0279 m

(a) Minati knows that light from the HeNe laser is fairly well collimated, so that nearly plane waves are incident on the pinhole, as illustrated in Figure 4-18b. She knows that plane waves are those that come—or appear to come—from very distant sources. So she concludes that, with the laser, the distance Z is much greater than 100 (0.0279 m)—that is, greater than about 2.8 m—and so the "Z-condition" for Fraunhofer diffraction is automatically satisfied.

(b) From her calculation of the ratio she knows also that the distance Z¢ must be greater than 2.8 m. So she can place the screen 3 meters or so from the aperture and form a Fraunhofer diffraction pattern—OR she can place a positive lens just beyond the aperture—as in Figure 4-18b—and focus the diffracting light on a screen a focal length away. With the focusing lens in place she obtains a much reduced—but valid—Fraunhofer diffraction pattern located nearer the aperture. She chooses to use the latter setup, with a positive lens of focal length 10 cm, enabling her to arrange the laser, pinhole, and screen, all on a convenient 2-meter optical bench.

2. Several typical Fraunhofer diffraction patterns. In successive order, we show the far-field diffraction pattern for a single slit (Figure 4-19), a circular aperture (Figure 4-20), and a rectangular aperture (Figure 4-21). Equations that describe the locations of the bright and dark fringes in the patterns accompany each figure.

Single Slit