Fresnel and Fraunhofer classes of diffraction

Fresnel Diffraction:

Fresnel diffraction occurs when the source of light, diffracting element, and observation screen are all at finite distances from each other. This is the case when the diffracting element is placed close to the source or the observation screen.

\[ I(\theta) = \left| \int \int U(x', y') e^{i\frac{2\pi}{\lambda z}(xx' + yy')} \, dx' \, dy' \right|^2 \]

Here,

• \( U(x', y') \) is the complex amplitude of the wave at the diffracting element,
• \( \lambda \) is the wavelength of the light,
• \( z \) is the distance between the diffracting element and the observation screen,
• \( (x, y) \) and \( (x', y') \) are coordinates in the observation plane and the diffracting plane, respectively.

Characteristics of Fresnel Diffraction:

1. Near-Field Pattern: Fresnel diffraction typically occurs when the distance from the diffracting element to the screen is comparable to the size of the diffracting element itself.
2. Complex Integral: The mathematical expression involves a double integral, making it more complex to calculate compared to Fraunhofer diffraction.
3. Wavefront Curvature: The curvature of the wavefront is taken into account in Fresnel diffraction, leading to more detailed patterns.

Fraunhofer Diffraction:

Fraunhofer diffraction occurs when the source, diffracting element, and observation screen are effectively at infinite distances from each other. This situation is achieved when the diffracting element is illuminated by parallel light (like light from a distant source).

\[ I(\theta) = \left| \int U(x', y') e^{i\frac{2\pi}{\lambda z}(xx' + yy')} \, dx' \, dy' \right|^2 \]

The difference here is that the expression is often simplified as the observation screen is far away (\( z \to \infty \)), leading to the use of Fourier transform relationships.

Characteristics of Fraunhofer Diffraction:

1. Far-Field Pattern: Fraunhofer diffraction occurs when the distance from the diffracting element to the observation screen is much larger than the size of the diffracting element.
2. Simpler Integral: The mathematical expression simplifies due to the far-field condition, making calculations more manageable.
3. Parallel Incident Light: Fraunhofer diffraction is often associated with parallel incident light, simulating light from a distant source.