Chapter 2: Wave Optics

Transcription

1 Chapter : Wave Optics P-1. We can write a plane wave with the z axis taken in the direction of the wave vector k as u(,) r t Acos tkzarg( A) As c /, T 1/ and k / we can rewrite the plane wave as t z u(,) r t Acos ( ) arg( A) T If the wave travels in the opposite direction to the z axis the wave is described by t z u(,) r t Acos ( ) arg( A) T When writing the wave in this way we describe the wave in any point and at any time. If we want to represent the wave by a diagram we must choose a particular value of t as when we take a picture of the wave. We also need to describe the variation of the optical amplitude with time at a certain point z on the wave to fully determine the wave. Determine A, T, and arg(a) for the optical wave that is described by the diagrams in figure -1. Determine also the direction of propagation, e.g. if the sign is positive or negative in the wave equation. Figure -1 In the diagram to the left we see the wave u(,) r t as a function of z when t = 0 s. To the right we see the optical wave amplitude at the origin as a function of t. Page 9

2 P-. An optical wave is described by t z u(,) r t Acos ( ) arg( A) T in air to the left and in a medium with refractive index n to the right. a) Determine by studying the diagrams in figure - values for A, T,, n and arg(a) and decide the sign + or. Do the same for both regions. b) Determine the phase vector that represents the complex amplitude U() r according to figure -, at the positions z =.0 m and z = 5.0 m at the time t = 0.0 s. Figure -. In the diagram to the left we see the wave u(,) r t as a function of z when t = 0 s. To the right we see the optical wave amplitude at the point z = 5.0 m as a function of t. P-3. Laser light is propagating in a transparent material. The electric field of the laser light varies according to E E0 sin( t kx) where E0 3.5 kv/m,.7 10 s and k m a) What is the wavelength of the laser light in the material? b) What is the refractive index of the material through which the light is propagating. c) What is the frequency of the laser light? * P-4. Validity of the Fresnel Approximation. Determine the maximum radius of a circle within which a spherical wave of wavelength = 633 nm, originating at a distance 1 m away, may be approximated by a paraboloidal wave. Determine the maximum angle m and the Fresnel number N F. Page 10

3 * P-8. A CD is illuminated with light from an Argon laser, see figure -6. The laser light consists of 6 visible lines of which the two strongest have the wavelength nm and nm. On a screen parallel with the laser ray a bright light spot is observed at a distance 373 mm (see the figure) and a bit from that position also 6 dots, of which the two strongest are at a distance 16 mm and mm (see the figure). All the dots are positioned at the same height above the laser table as the incident laser ray. Calculate the distance between the tracks on the disc. Make the calculations for both wavelengths. Laser CD 00 mm 300 mm Screen mm 373 mm Figure -6. Observation of diffraction spots from a CD. P-9. The determination of the wavelength of X-ray radiation is performed by using a reflection grating. If light with the wavelength nm from a Cd-lamp is impinging perpendicularly against the grating, the first order is observed at a deflection angle of º. To reflect X-ray the light has to be at grazing incidence according to figure -7. The zero order beam (m = 0) is observed at D and the first order is observed at E. Calculate the wavelength of the X-ray radiation. The distance d between grating and screen is 1.0 m. Figure -7. X-ray diffraction from a grating. Page 11

4 * P-10. The light from the slit in a spectrometer is collimated with a lens with the focal length.00 m. The light meets a reflection grating at the incidence angle 19.00º. The returning light is focused with the same lens and in the slit plane there is a photographic plate. See figure -8. The grating has 1500 grooves/mm. a) At which angle is the first order reflected from light with the wavelength nm? b) Another spectral line is also observed 45 m from the line in problem a). Calculate the difference in wavelength between the two lines. Light source Fotographic film Lens 19 o Reflection grating Figure -8. Reflection of light in a spectrometer. P-11. Bragg Reflection. Light is reflected at an angle from M parallel reflecting planes separated by a distance d as shown in figure -9. Assume that only a small fraction of the light is reflected from each plane, so that the amplitudes of the M reflected waves are approximately equal. Show that the reflected waves have a phase difference k(dsin ) and that the angle at which the intensity of the total reflected light is maximum satisfies sin (.5-11) d This angle is known as the Bragg angle. Such reflections are encountered when x-ray waves are reflected from atomic planes in crystalline structures. It also occurs when light is reflected from a periodic structure created by an acoustic wave. Figure -9. Reflection of a plane wave from M planes separated from each other by a distance d. The reflected waves interfere constructively and yield maximum intensity when the angle is the Bragg angle. Page 1

5 * P-1. In a salt crystal (according to figure -10) the deviation (angle between the transmitted and Bragg reflected light) of two X-ray wavelengths is 6.30 and The distance between close crystal planes in the NaCl crystal is 8 pm. Calculate the two X-ray wavelengths if you assume that it is the first order Bragg reflection that causes the deviation. Figure -10. Sodium chloride is crystallized in a cubic grating. At the corners of the small cubes a sodium or a chloride atom is situated. In the figure the size of the atoms is too small in relation to the distance between them. * P-13. Yellow light from a Sodium light source consists of two frequencies. In air the wavelengths are nm and nm. How long shall the distance between the mirrors in a Fabry-Perot interferometer be so that one misses to see that the light consists of two frequencies? Hint! Assume that one ring with order number m from one wavelength coincides with the ring with order number m+1 from the other wavelength. You can also calculate the frequency difference between the two wavelengths and use the relation f c/d. Figure -11. Interference pattern observed from two point sources. P-14. Interference of Two Spherical Waves. Two spherical waves of equal intensity I 0 originates at the points (a,0,0) and (-a,0,0) interfere in the plane z = d as illustrated in figure The system is similar to that used by Thomas Young in his celebrated double-slit experiment in which he demonstrated interference. Use the paraboloidal approximation for the spherical waves to show that the detected intensity is x I( x, y, d) I0 1cos. (FoP.5-8) where a/ d is approximately the angle subtended by the centers of the two waves at the observation plane. The intensity pattern is periodic with period /. Page 13

6 * P-15. Two coherent equally intense laser beams with the wavelength 53 nm intersect with an angle according to the figure -1. In the region of overlap an interference pattern is obtained. This pattern consists of parallel planes with either high or low intensity. Calculate the distance between two nearby planes of high intensity. This is the principle of making holographic gratings. The method of producing a reflecting grating is the following. A plane glass surface is covered with photoresist. After exposure and development (where the exposed resist is removed) the surface is covered with a thin reflective layer. a Figure -1. Inteference pattern observed where two coherent beams overlap. * P-16. Fringe visibility. The visibility of an interference pattern such as that described by (FoP.5-4) and plotted in FoP, Figure.5-1, is defined as the ratio: V ( Imax Imin )/( Imax Imin ) where I max and I min are the maximum and minimum values of I. Derive an expression for V as a function of the ratio r I1/ I of the two interfering waves and determine the ratio I1/ I for which the visibility is maximum. Page 14

7 * P-17. A schematic of the Michelson interferometer is shown in figure -13a. The light (1) from an extended light source (a transparent screen illuminated by a light source) is split by a beam splitter BS into two beams () and (3). The beams are reflected at the mirrors M1 and M and returned to the beam splitter. In the beam splitter two beams are created, one that goes back to the light source and one (4) that is directed to the observer. If the light source is a white lamp with a broad spectrum it is a necessity to use a compensator plate C to observe fringes. If the interferometer is correctly adjusted a circular ring system is observed localized at infinity. The fringes are due to the interference of two images of the light source as shown in figure -13b. An off axis point P will thus be imaged as two points P 1 and P separated by a distance d where d is the difference in distance that is due to a different position of the mirrors. In the following we assume that we use a HeNe light with the wave length 633 nm. a) Derive a relation between the optical path difference p as a function of the angle and distance d. b) We adjust the interferometer so that the path difference between the two paths is zero. After that we move one mirror a distance of 4.0 mm and adjust slightly to observe a dark area in the center. At which angle do we observe the nearby dark ring? c) A gas cell is now placed in one of the arms of the interferometer. The length of the gas cell is 100 mm. We start with vacuum and we fill the gas cell with gas. After we have stopped filling with gas, we find that 155 central interference fringes have passed. Calculate the refractive index of the gas. M p P P 1 S S1 d S () C M M1 d (1) BS (3) (4) M1 S P Figure -13a. Schematic view of a Michelson interferometer. Figure -13b An equivalent drawing for the rays in the Michelson interferometer in figure -13a. * P-18. Graded-index lens. (Exercise.4-6, Fundamentals of Photonics) Show that a thin plate of uniform thickness d 0 (Fig FoP) and quadratically graded 1 refractive index nx, y n0 1 x y, with d 0 1, acts as a lens of focal length f 1/ n d (also see exercise FoP). 0 0 Page 15

8 Answers chapter : P-1: A 1.5 mv/m, T = 0ns, = 6 m, arg (A) = 3,+,since the wave travels to the left. P-: a) A 3.5 mv/m, T = 0ns, = 6 m, arg(a) = 3,,since the wave travels to the right. In the right region we have A.8 mv, T = 0 ns, = 4. m, n = 1.43, arg(a) =1.3 j4 /3 b) u 3.5 e mv/m P-3: a) 488 nm b) n = 1,70 c) Hz P-4: a << 4 cm, 0,04 rad, N 514 P-8: 1.6 μm P-9: 0.18 nm P-10: a) b) 13.9 pm P-1: 18 pm and 144 pm P-13: 0.9 mm m F P-16: V r ; I 1 / I 1 r 1 P-17: a) p dcos b) 0.7 c) n = Page 16