3.3 Fraunhofer diffraction: far-field patterns

Fraunhofer diffraction occurs in the far-field region, where light waves are treated as plane waves. This simplifies calculations and allows for the use of Fourier transforms to analyze diffraction patterns. The conditions for Fraunhofer diffraction include plane wave incidence and large observation distances.

The diffraction pattern depends on the aperture's shape and size. Rectangular apertures produce sinc functions, while circular apertures create Airy patterns. Increasing aperture size improves resolution and brightness but adds more secondary maxima to the diffraction pattern.

Fraunhofer Diffraction: Far-Field Patterns

Conditions for Fraunhofer diffraction

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• Incident light must be a plane wave or spherical wave from a point source at infinity enables simplified mathematical analysis
• Diffracted light observed at a large distance from the aperture (far-field region) allows for the use of Fourier transform in calculations
• Aperture size much larger than the wavelength of incident light ensures the validity of scalar diffraction theory
• In the far-field region, diffraction pattern independent of distance between aperture and observation plane simplifies calculations
• Fraunhofer diffraction relevant in far-field region allows analysis of diffraction pattern using Fourier transform of aperture function (telescopes, microscopes)

Mathematical derivation of diffraction patterns

• Complex amplitude of diffracted light in far-field region proportional to Fourier transform of aperture function $U(x, y)$ enables mathematical analysis
• Fraunhofer diffraction pattern given by:
  • $U(x, y) = \frac{e^{ikz}}{i\lambda z} e^{ik\frac{x^2+y^2}{2z}} \iint_{aperture} U(x', y') e^{-i\frac{2\pi}{\lambda z}(xx'+yy')} dx' dy'$
    • $k = \frac{2\pi}{\lambda}$ wave number relates wavelength to spatial frequency
    • $\lambda$ wavelength of incident light determines scale of diffraction pattern
    • $z$ distance between aperture and observation plane affects size of diffraction pattern
    • $x', y'$ coordinates in aperture plane define the shape of the aperture
    • $x, y$ coordinates in observation plane determine the location of the diffraction pattern

Diffraction patterns of aperture geometries

• Rectangular aperture:
  • Aperture function for rectangular aperture of width $a$ and height $b$: $U(x', y') = rect(\frac{x'}{a}) rect(\frac{y'}{b})$ describes the shape of the aperture
  • Fraunhofer diffraction pattern for rectangular aperture: $U(x, y) = \frac{e^{ikz}}{i\lambda z} e^{ik\frac{x^2+y^2}{2z}} ab \, sinc(\frac{a}{\lambda z}x) sinc(\frac{b}{\lambda z}y)$
    • $sinc(x) = \frac{sin(\pi x)}{\pi x}$ normalized sinc function determines the shape of the diffraction pattern
• Circular aperture:
  • Aperture function for circular aperture of radius $a$: $U(r') = circ(\frac{r'}{a})$, where $r' = \sqrt{x'^2 + y'^2}$ describes the shape of the aperture
  • Fraunhofer diffraction pattern for circular aperture: $U(r) = \frac{e^{ikz}}{i\lambda z} e^{ik\frac{r^2}{2z}} \pi a^2 \, \frac{2J_1(\frac{ka}{\lambda z}r)}{\frac{ka}{\lambda z}r}$
    • $J_1(x)$ first-order Bessel function of the first kind determines the shape of the diffraction pattern
    • $r = \sqrt{x^2 + y^2}$ radial coordinate in observation plane defines the location of the diffraction pattern

Effects of aperture on diffraction

• Aperture size effects:
  1. Increasing aperture size decreases width of central maximum in diffraction pattern (higher resolution)
  2. Increasing aperture size increases intensity of central maximum (brighter image)
  3. Increasing aperture size increases number of observable secondary maxima (more detail in diffraction pattern)
• Aperture shape effects:
  • Shape of aperture determines overall shape of diffraction pattern (rectangular vs. circular)
  • Rectangular apertures produce diffraction pattern with central maximum and secondary maxima along x and y axes (cross-shaped pattern)
  • Circular apertures produce diffraction pattern with central maximum surrounded by concentric rings (Airy pattern)
  • Relative intensity and spacing of secondary maxima depend on specific shape of aperture (number and distribution of rings/maxima)