# Pedrotti and Pedrotti Introduction to Optics - Sample Essay

Fourier optics is seen as an extension of Huygens-Fresnel principle. It is the study of classical optics that involves the Fourier transforms. The ideology behind this technique is that any wave that is moving has an infinite number of wave points which after colliding with an obstacle on the way may be seen to take different directions from the original trajectory indicating that this wave points are independent from each other i. e. a Franhoffer diffraction can be established from Fourier transforms. In Fourier transform technique when the wave is close enough it is very much possible to pay attention to the individual wave points.

Using mathematical calculations in this step fundamentally explains the basis of Fourier analysis and synthesis. The Fourier analysis and synthesis normally explains the principles that are involved when light waves pass through various slits or mirrors is fully or partially reflected or curved one way or the other. This technique forms the basis of the concepts that are involved in image processing techniques as well as its fundamental applications in light wave situations. Fourier optics is used mostly as the conjugate of the spatial (x,y) domain for spatial frequency domain (kx, ky).

In Fourier transform techniques, terminologies and concepts such as spectrum, bandwidth, transform theory, window functions and sampling from one dimensional signal processing are used frequently. Materials and methods 1- A filter holder was placed in front of the beam expander and a 2-slit pattern selected. A 2-slit diffraction pattern was observed from a distance. 2- The simple screen was replaced with the photo-diode detector assembly that was provided. This was detector translated across the pattern and the intensity distribution was captured by the software package. A software package called Science Workshop was used.

3- A lens was placed approximately 1 focal length beyond the slit system. The screen was moved until an image of the 2 slits was observed. At this stage the lens obeyed the lens equation 1/s +1/s’ = 1/f where s, sā are object and image distances respectively and f is the focal length of the lens. 4- Re-positioning was done for the screen so that it was 1 focal length beyond the lens and it was observed that the far-field (Fraunhofer) diffraction pattern of the 2 slits is now appeared. 5 ā A second lens was placed 1 focal length beyond the transform plane in order to reinforce the idea that a lens converts from real space to Fourier space.

An examination of the image formed 1 focal length beyond the second lens was done and it was confirmed that it was indeed an image of the real slit system. 6 – A number of slit systems were provided so that an exploration of their diffraction patterns could be done in detail. The intensity patterns of a number of patterns in both diffraction and real space were recorded using the Science Workshop software package. 7- Excel software package was used to calculate the Fourier transform of a 4-slit system. Comparisons were made between the 4-slit diffraction pattern with the graphical result of the Fourier transform.

Variations were done on the slit width / slit separation ratio and the effects on the associated diffraction pattern was examined. 9- As an introduction to image filtering, a variable aperture was placed at the transform plane of the 1st lens when the 4-slit pattern being used.

References

Pedrotti and Pedrotti Introduction to Optics (2nd Edition) Prentice Hall 1993 Chapter 25. Hecht Optics (2nd Edition) Addison ā Wesley 1987 Chapter 11 and Section 14. 1. Scott, Craig (1998). Introduction to Optics and Optical Imaging. Wiley.