3.2 Fresnel diffraction: near-field effects

Fresnel diffraction occurs when light waves are observed close to their source. This near-field effect creates complex patterns that depend on distance and aperture shape. Understanding Fresnel diffraction is crucial for analyzing light behavior in optical systems.

Calculating Fresnel patterns involves complex integrals and special functions. As the observation distance increases, Fresnel diffraction gradually transitions to Fraunhofer diffraction. This concept is vital for applications like near-field microscopy and zone plate design.

Fresnel Diffraction and Near-Field Effects

Characteristics of Fresnel diffraction

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• Occurs when observation distance is comparable to aperture size and light wavelength
  • Near-field region defined as $z \ll \frac{a^2}{\lambda}$, $z$ is observation distance, $a$ is aperture size, $\lambda$ is wavelength
• Exhibits complex structure with bright and dark fringes
• Pattern not a simple Fourier transform of aperture shape
• Depends on distance between aperture and observation plane
• Wavefront curvature cannot be neglected in near-field region

Calculation of Fresnel diffraction patterns

• Fresnel diffraction integral $U(x,y) = \frac{e^{ikz}}{i\lambda z} \iint_\Sigma U_0(\xi, \eta) e^{\frac{ik}{2z}[(x-\xi)^2 + (y-\eta)^2]} d\xi d\eta$
  • $U(x,y)$ complex amplitude at observation point $(x,y)$
  • $U_0(\xi, \eta)$ complex amplitude at aperture
  • $k = \frac{2\pi}{\lambda}$ wavenumber
• Fresnel integrals $C(v) = \int_0^v \cos(\frac{\pi}{2}t^2) dt$ and $S(v) = \int_0^v \sin(\frac{\pi}{2}t^2) dt$
  • Calculate diffraction pattern for simple apertures (rectangular, circular)
• Rectangular aperture $U(x,y) = \frac{U_0}{2} \left[ \left(C(v_2) - C(v_1)\right) + i\left(S(v_2) - S(v_1)\right) \right]$
  • $v_1 = \sqrt{\frac{2}{\lambda z}}(x - \frac{a}{2})$ and $v_2 = \sqrt{\frac{2}{\lambda z}}(x + \frac{a}{2})$, $a$ is aperture width
• Circular aperture $U(r) = U_0 \left[ \frac{1}{2} - C(v) + i\left(\frac{1}{2} - S(v)\right) \right]$
  • $v = \sqrt{\frac{2}{\lambda z}}r$, $r$ is radial distance from diffraction pattern center

Transition from Fresnel to Fraunhofer

• Fraunhofer diffraction occurs when observation distance much larger than aperture size and wavelength ($z \gg \frac{a^2}{\lambda}$)
• As observation distance increases, Fresnel pattern gradually transitions to Fraunhofer pattern
  • Wavefront curvature becomes less significant
  • Diffraction pattern becomes Fourier transform of aperture shape
• Transition characterized by Fresnel number $N_F = \frac{a^2}{\lambda z}$
  • Fresnel diffraction dominates when $N_F \gg 1$
  • Fraunhofer diffraction dominates when $N_F \ll 1$
  • Transition occurs when $N_F \approx 1$

Applications of Fresnel diffraction theory

• Analyze near-field diffraction phenomena in optical systems
  • Diffraction by apertures and obstacles in near-field region
  • Light propagation through optical components (lenses, mirrors) in near-field region
  • Near-field imaging and lithography techniques
• Fresnel zone plate
  • Alternating transparent and opaque concentric rings
  • Focuses light by diffraction in near-field region
  • Focal length depends on wavelength and zone plate geometry
• Near-field scanning optical microscopy (NSOM)
  • Uses subwavelength aperture or tip to probe sample's near-field region
  • Achieves resolution beyond diffraction limit by exploiting evanescent waves in near-field region
• Crucial for designing and optimizing near-field optical systems