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Fresnel and Fraunhofer diffraction

This application can be used to analize the intensity of the diffracted field in Fresnel or Fraunhofer conditions for some remarkable apertures. The program supposes plane wave illumination normal to the plane that contains the aperture. Under Fresnel conditions, the applet calculates the diffraction for rectangular and circular apertures, slits and the half-plane, while under Fraunhofer conditions, the calculation is carried out for rectangles, circles and slits.

Fresnel diffraction

Let U(x,y,z) be the diffracted field propagated at a distance z. The plane that contains the aperture Σ is located at z=0. U(x₀,y₀,0) is the field in the plane z=0 and (x₀,y₀) are the coordinates in this plane. k is the wave number and λ is the wavelength.

The calculation under Fresnel conditions can be carried out when the differences x-x₀ or y-y₀ are smaller than z. In this case, the equation that relates the diffracted field U(x,y,z) with U(x₀,y₀,0) is:

where the intensity is calculated as the square of the modulus of the diffracted field:

The program calculates this integral in different situations. The user should introduce the following data:

1. The wavelength, in the range of 400 to 700 nm.
2. The propagation distance z, among 100 and 1000 mm.
3. The object that the light passes through:
   • rectangle: the values of both sides, between 0.5 and 3 mm
   • circle: the diameter varies between 0.5 and 3 mm
   • slit: the width varies between 0.5 and 3 mm
   • half-plane

Next, the button "Calculate diffraction" should be pressed. After a while, the intensity of the diffraction will be shown in the right side of the applet window. This image can be processed in order to analyze the areas of the image with smaller energy. The user can choose among the following visualization options:

1. Intensity (default option).
2. The logarithm of the intensity.
3. Amplitude (the square root of the intensity). When selecting this option the "Threshold" slider is activated. The maximum value of the amplitude of the diffraction is initially normalized to 100. When selecting a certain value of the threshold, all the values above it are adjusted to 100 and the values below the threshold are renormalized from 0 to 100.

The profile of the intensity of the y=0 axis is shown when pressing on the button "Central line profile". The plot corresponds to the intensity of the diffraction without modifications.

Fraunhofer diffraction

The calculation under Fraunhofer conditions is carried out when the propagation distance z tends to infinite. In this case, the equation that relates the diffracted field U(x,y,z) with the field in the aperture U(x₀,y₀,0) is:

In practice, a convergent lens of focal f' can be used to observe the intensity of the diffracted field in Fraunhofer conditions. The aperture is illuminated with a plane wave and then the diffracted field passes through the lens. It can be demonstrated that the electric field obtained in the focal plane of the lens is equivalent to the field propagated a distance z that tends to infinite:

where the intensity is calculated as the square of the modulus of the diffracted field.

The program calculates this integral in different situations. The user should introduce the following data:

1. The wavelength, in the range 400 to 700 nm.
2. The image focal of the lens between 500 and 1000 mm.
3. The object that the light passes through:
   • rectangle: the values of both sides between 0.5 and 3 mm
   • circle: the diameter varies between 0.5 and 3 mm
   • slit: the width varies between 0.5 and 3 mm

   These objects can be replied up to ten times. The distance among them is constant.

The image can be processed in order to analyze the areas of the image with smaller energy. The user can choose among the following visualization options:

1. Intensity (default option).
2. The logarithm of the intensity.
3. Amplitude (the square root of the intensity). When selecting this option the "Threshold" slider is activated. The maximum value of the amplitude of the diffraction is normalized initially to 100. When selecting a certain value of the threshold, all the values above it are adjusted to 100 and the values below the threshold are renormalized from 0 to 100.

The profile of the intensity of the y=0 axis is shown when pressing the button "Central line profile". The plot corresponds to the intensity of the diffraction without modifications.