Chapter 4: Fourier Optics

Transcription

1 Chapter 4: Fourier Optics P4-1. Calculate the Fourier transform of the function rect(2x)rect(/3) The rectangular function rect(x) is given b 1 x 1/2 rect( x) when 0 x 1/2 P4-2. Assume that ( gx (, )) G ( x, ) e.g. the function G is the fourier transform of g. a) Calculate ( gx ( 0.4, 2.3)) b) For which values of x and is the following true: ( gx ( 0.4, 2.3)) G (, ) x * P4-3. Assume that the rectangle in figure 3-1a below has a length of c (bottom side). It is placed in front of a lens with a focal length f. Calculate the distance to the first horizontal minimum in the Fourier transform of the object b: a) using the two figures (Figure 3-1b) below. b) using ordinar diffraction relations e.g. building up the image from b adding waves. Figure 3-1a. A rectangular opening Figure 3-1b. A function and its Fourier transform. P4-4. A transparenc with the transmission function t(x,) is placed close to a lens with a focal length of 20 cm. How far from the optical axis is it possible to observe the vertical spatial frequenc 30 ccles/mm, if the wavelength of the light is 633 nm. Page 17

2 * P4-5. Using the figure 3-2 below, derive the following formula: f w0, w which gives the beam waist 2w 0 when a Gaussian beams of width 2w is focused b a lens with the focal length f. The beam waist is the diameter of the beam defined where the electric field amplitude has dropped to 1/e of the top value. Figure 3-2. A Gaussian function and its Fourier transform. * P4-8. In the next page ou see several figures or images. a) Combine six of the figures (figures a i below) two and two (object and FT) and describe how the three remaining figures are related. Motivate! b) Describe how and in which figure linearit of the Fourier transform is clearl seen. c) Describe how convolution can be performed using Fourier transforms? a b c d e f g h i Figure 3-3. Different objects and Fourier transforms. Page 18

3 * P4-9. The optical processor shown in figure 3-4a is used for spatial filtering. A cross grating according to figure 3-4b is the object. The result of a vertical and horizontal spatial filtering of the object is shown in figure 3-4c. Assume now that we insert a narrow slit at a 45 angle in the transform plane of figure 3-4a below. How will the spatiall filtered image of the object in figure 3-4c change? Plane wave Object plane Transform plane x L 1 x L 2 Image plane f 1 f 1 f 2 x The Fourier transform of the letter E f 2 Figure 3-4a An optical processor Figure 3-4b A cross grating and its Fourier transform. Figure 3-4c. Vertical and horizontal spatial filtering. P4-10. Assume that a slide with the amplitude transmission function given as g( x) cos2 0x is placed in an imaging sstem with the cut-off frequenc Is it better with coherent illumination than with incoherent? n 2 1 j2 n2 0x Hint: cos2 0x e 2 n 1 2n Draw the spectrum and the transfer functions. Hint! For a coherent sstem the sstem is linear for amplitudes. The transfer function is given b the function H ( x, ) as shown in FoP figure For an incoherent sstem the transfer function is H* H which has the double cut-off frequenc Note also that the transfer function operates on the intensit of the image instead of the amplitude. Page 19

4 P4-11. The first minimum in the Air pattern is given b equation 4.3-8: 1.22 D The second minimum is given b: 2.23 D The following minima can be described with: k D where k = m with integers m 3. To determine the diameter of a circular hole it is illuminated b red light from a He-Ne laser (λ = nm). The diffraction pattern is studied on a screen at a distance of 5.00 m from the hole. The diameter of the fifth dark ring as measured from the bright central spot is 62 mm. Calculate the diameter of the hole. * P4-12. Figure 3-5a below shows a bubble chamber photograph and figure 3-5b is a filtered version of the same image. Figure 3-5c illustrates the Fourier Transform of figure 3-5a. a) Show in a figure what the Fourier transform of figure 3-5b would look like. Indicate the main difference compared to figure 3-5c. b) Describe in a figure what the spatial filter should look like in order to obtain the filtered image in figure 3-5b. c) With the lens L4 removed, the Fourier transform of the original image (figure 3-5a) is observed on a screen (figure 3-5d). Calculate the distance (on the screen) from the optical axis to the first vertical minimum in the Fourier transform of the horizontal lines (average width of the lines is 0.24 mm ) if the image was placed 9 cm after L2 as shown in figure 3-5d. The wavelength of the light is 633 nm. Note that the scale of the Fourier transform is determined b the distance from the object to the Fourier plane. d) If the lens L4 is introduced in the set-up, an enlarged image of the object is obtained at the screen. Calculate the focal length of the lens L4 that will make this image sharp. Figure 3-5a Figure 3-5b Figure 3-5c L1 L2: f = 38 cm L4 156 cm 9 cm 13 cm 24 cm Screen L3:f=12 cm Figure 3-5d. Optical set-up for observation of Fourier transforms and spatial filtering. Page 20

5 P4-13. A hologram can, in some cases, have a resolution of 5000 lines/mm. How man values of photographic densit must be known to completel describe a hologram with an area of 1 mm 2? * P4-14. A photographic plate C used to create a hologram (figure 3-6) is illuminated with light from two coherent light sources A and B. The wavelength of the light is m. a) What will be the distance between nearb fringes in the point P? b) When the plate is developed a hologram is formed. This means that in P ou have locall a grating with the grating constant given from a). This hologram is mounted in the same position as before development and illuminated with laser light coming onl from the source in A. Calculate the direction for which ou observe light on the other side of the plate coming from the point P mm A 1000 mm 500 mm C 500 mm A C B P B 100 mm P Figure 3-6 a) Two point sources (A and B) illuminates a photographic plate C. c) The illumination of the plate (hologram) after development. Page 21

6 Answers chapter 4: P4-1:1.5sinc( f x / 2) sinc(3 f ) P4-2: a) G( fx, f) expj2 ( fx0.4 f 2.3) b) ( fx0.4 f 2.3) 0, 1, 2,... P4-3: x f / c P4-4: 4 mm P4-6:a) x 2 : 0, 2 x 1: 4 2 x, 1 x 0 : 5 3 x,0 x 1: 5 3 x,1 x 2 : 4 2 x. b) x<0; 0, 0 x 1: x,1 x 2 : 3x2,2 x 3: 4,3 x 4 :16 4x. x x P4-7:( x) when -1 < x < 1, ( x) = 0 when x P4-8: a) a-f, c-d, e-i, b is the correlation of h and g b) c is a sum of two object: a triangle and a pattern of small openings. d is the sum of the FT of a triangle and the FT of the openings c) First take the FT of each function. Multipl the result and take IFT of the new result. This is the convolution of the original functions. This seams complicated but is done quickl b a FFT(Fast Fourier Transform) algorithm. P4-9: The lines will lean 45 and the distance will change to 1/ 2 times what it was before. P4-10:The incoherent sstem is the best. P4-11: 0.54 mm P4-12 a) b) FT of figure 3-5a FT of figure 3-b Opening for the zero sp. frequenc P4-13: c) 9 mm d) f = 6.8 cm 8 10 From lines in figure 3-5a. P4-14: a) 1.52 μm b) 68, 76,31.05,5.71, and Page 22