Fourier Optics The methods of harmonic analysis have proven to be useful in describing signals in many disciplines. In this experiment you will explore the concepts of Fourier analysis as applied to the field of optical imaging. propagation of Fourier light optics waves provides based on a description harmonic analysis of the (Fourier decomposition). Harmonic analysis is based on the expansion of an arbitrary function () as a superposition (a sum or an integral) of harmonic functions of different spatial frequencies, . The harmonic function ( ) − , which has a spatial frequency and an amplitude ( ) is the building block of the theory. Several of those functions, each with its own spatial frequency and amplitude are added to construct the function (). The function ( ) is called the Fourier transform of (). The two functions are related by the following Fourier transformations: +∞ () = ∫ −∞ ( ) = ∫ ( ) − +∞ () + −∞ In this experiment you will learn how to decompose an object, () into its Fourier components described by a certain function ( ) − , propagate those spatial frequencies through an optical system composed of multiple lenses, and create a magnified image of the object, (), and an image of the Fourier transform ( ) of the object. You will also measure the spatial frequencies present in an 1 object and determine relevant feature sizes of the object. You will learn how to filter spatial frequencies and verify its impact in the image of an object. Experiment: A He-Ne laser beam at 0 = 632.8 nm, which out of the laser head typically has a beam diameter of 1 mm, has been expanded to about 20-30 mm in diameter by a combination of two lenses, one with a short focal length (a microscope objective) and another one with a long focal length. The distance between the two lenses has been adjusted to create an expanded optical beam with a flat wavefront (i.e., an expanded collimated beam). The flat wavefront is confirmed by a shear plate interferometer. Next, the light wave with a large diameter size and a flat wavefront is ready to illuminate objects of interest. As objects to be illuminated in this experiment you will use several meshes with different grid sizes. Start with the mesh that is marked with 120, which is supposed to have a grid size of 120 µm according to the vendor. Place the mesh at about 100 mm in front of the first lens of the optical system that will come after the object. The optical system placed after the object is illustrated in the Figure below and is composed of two lenses, each with a focal length of = 100 and they are 2 separated by about 200 mm, which is 2 = 200 . When the object is placed at about the object focal point of the first lens, 100 , and image of the object will appear at the image focal point at about 100 of the second lens. This configuration is known in the literature as 4 system. At a distance from lens 1, each spatial frequency coming out of the object will converge to a point. This plane is known as the Fourier plane. Put a white piece of paper to observe the light pattern at this plane. Now, as light propagates through the optical system, lens 2 sees an object (the Fourier pattern) at its object focal plane and will form an image of this pattern at a screen at infinity (at very large distance from it). You will take measurements to describe the Fourier pattern at a 3 screen at a large distance from lens 2. Measure the distance form lens 2 and the screen used for observation, . The transverse magnification given by || = of the Fourier pattern is then ( −) Sketch a drawing in your notebook of the pattern you observe at the screen. Consider the brightest spot in the pattern as a reference point and take measurements for the location of the several other bright spots you observe in image of the Fourier pattern at the screen. Bring a new lens (lens 3) to the setup to create a magnified image of the real object at the screen. Find the location to form a sharp image of the object on the distant screen. Move the lens 3 in and out of the optical path. You will observe the magnified object pattern (, ) when the lens is in the optical path, and the magnified Fourier pattern ( , ) when the lens is out of the optical path. You are performing a two dimensional Fourier transformation by moving the lens in and out of the optical path. Insert a slit (with the blades oriented in the vertical direction) at approximately the Fourier plane. Because the opening of the slit is set along the vertical direction, it will remove horizontal some spatial direction, . frequencies Slowly move spread the slit in in the the horizontal direction and observe what happen at the screen. Bring lens 3 in and out of the optical path. You should be 4 able to remove the horizontal lines from the magnified image of your object. Analysis: Consider the bright spots in one line in the vertical direction. The spatial frequencies in the vertical direction will then be given by: (, ) = ∆ ⁄ , where ∆ is the verticalistance to the reference point for each bright spot marked by integer = ±1, ±2, ±3, … and the positive and negative signs correspond to above and below the reference point, respectively. As the image at the distant screen of the Fourier pattern has been magnified, the corresponding spatial frequency , in the back focal plane of lens 1 (Fourier plane in the Figure) is given by: , = (, ) = 2 0 (, ). Given a mesh with a regular vertical grid size of Λ , the spatial frequency in the vertical direction will be given by , = 2 Λ = 2 0 (, ). Create a table of and = 1 0 1 0 (, ). Plot = versus (, ). The slope of a linear fit of versus should give you the vertical grid size Λ . Repeat your analysis above for the horizontal direction and determine Λ . 5 Questions to consider: 1. What would happen if the grid size was not constant along the illuminated pattern? 2. What would happen if Λ was smaller than the laser wavelength? 3. Mathematically describe the two dimensional object with a regular square pattern along in x-y plane. Calculate the Fourier transformation of this two dimensional pattern. 6 https://utahbiodieselsupply.com/stainlessmesh.php 7

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