Main Page/PHYS 4210/Fourier Optics

Fourier Optics

In this experiment we investigate the diffractive (as opposed to refractive) properties of a lens, named Fraunhofer diffraction. A HeNe laser is used to illuminate an image or a grating whose Fourier image is generated in the focal plane of a lens. The Fourier image is modified by cutting away low or high components (orders of the diffraction pattern) and a second lens is used to view the altered image.

Introduction

Ray optics deals with the simple design of optical systems. While trying to take, e.g., a microscope to its limit, i.e., to improve the resolution, researchers in the middle of the 19th century (Abbé, in particular) found that the dark parts of the image inside an optical system contribute to the final image. Physical optics makes use of the wave nature of light to understand these phenomena. It turns out that a simple lens produces in its focal plane a diffraction pattern for the image. In ref. 1 detailed explanations of the mathematical description of imaging are provided.

The problem is summarized in Fig. 1, which shows how parallel light illuminates an object, and a symmetric set-up of two high-quality lenses. The first lens is used to construct a Fourier image in the focal plane, while the second lens forms back the image. The diffraction pattern in the focal plane (the drawing is rather artistic, i.e., inaccurate) is easily observed for monochromatic light (a mercury lamp with a colour filter, or a laser), and can be modified (filtered) to modify the image.

The description of the pattern in the focal plane becomes straightforward if we consider a simple image, such as, e.g., a grating. From previous optics demonstrations you may be familiar with two types of gratings that are characterized by their line density (in lines/inch): (i) simple rulings (e.g., Ronchi rulings for the measurement of the resolution of lenses and their spherical aberration) have an intensity profile that corresponds to a square-wave pattern; these rulings when illuminated by a monochromatic source generate a diffraction pattern of many equidistant dots along a line perpendicular to the rulings; (ii) gratings with a sinusoidal intensity patterns are often used to produce a single pair of diffraction maxima. This corresponds to the Fourier representations of a sinusoidal intensity pattern: a single cosine mode (central spot) and two sine-terms, while the Fourier series for case (i) has infinitely many contributions.

The location of the diffraction maxima depends on the spacing of the ruling d and the wavelength λ of the light. A useful parameter that appears in the intensity patterns of apertures is given by