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Wave Optics Chapter 27 Wave Optics Linear Superposition and Interference The Double-Slit Experiment Interference in Thin Films Interferometers Interference and Diffraction Diffraction and Spatial Resolution Diffractive Spectral Dispersion: Gratings Linear Superposition and Interference From Chapter 17: The Principle of Linear Superposition When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves. In the chapter 17 context, we were talking about mechanical waves. But the principle holds for electromagnetic waves as well. Linear Superposition and Interference A few terms: Coherent sources of light emit waves that have a constant phase relationship. Constructive interference occurs when the field vectors have the same condition at the same place at the same time. Destructive interference occurs when the field vectors are exactly opposite in condition at the same place at the same time. Linear Superposition and Interference If two waves from a coherent source travel two different paths to arrive at a common point, their phase relationship at that point depends on the difference in lengths between their paths. Linear Superposition and Interference Condition for destructive interference: 1 path m 2 Condition for constructive interference: path m (where m is any integer) Young’s Double-Slit Experiment Image courtesy of University of Toronto physics department Thomas Young, 1773 – 1829 English physicist, medical doctor, and Egyptologist Also the inventor of Young’s modulus (strength of materials) Young’s Double-Slit Experiment Young’s experimental setup: Young’s Double-Slit Experiment Young’s result: Young’s Double-Slit Experiment Path difference geometry: Young’s Double-Slit Experiment Conditions for dark and bright fringes: m bright fringes: sin d 1 m 2 sin dark fringes: d Thin-Film Interference For light, as for any electromagnetic wave, wavelength, frequency, and speed are related by c f This applies in vacuum, where the speed is c. In a material with refractive index n, c c n n f nf n The wavelength has been reduced by a factor of n. Thin-Film Interference Two glass plates in air: 2 1 t Wave 1 is reflected from the lower surface of the upper plate Wave 2 is reflected from the upper surface of the lower plate Wave 2 travels 2t farther than does wave 1 Thin-Film Interference But we also find out … 2 1 t … that when a wave is reflected while trying to pass from a lower index into a higher one, it experiences a change of phase at the interface that is equivalent to 1/2 in the higherindex material: n / 2. Thin-Film Interference How many “extra” wavelengths … 2 1 t … does wave 2 experience, relative to wave 1? 2t # of extra wavelengths: extra distance 1 2 phase change Thin-Film Interference Condition for constructive interference: # of wavelengths = m 2 1 t 2t 1 m 2 1 1 t m 2 2 Thin-Film Interference 2 1 1 t m 2 2 1 t We see another bright fringe (constructive interference) for every half-wavelength change in the thickness of the air wedge. Thin-Film Interference For destructive interference: 1 # of extra wavelengths = m 2 2 1 t 2t 1 1 1 m t m 2 2 2 A dark fringe also occurs at half-wavelength intervals of air wedge thickness. Thin-Film Interference Consider a soap bubble. Condition for constructive interference between ray that reflects from the outer surface and ray that reflects from the inner surface: 1 2t film m film 2 1 1 1 1 t m film m 2 2 2n film 2 Destructive interference: m t 2n film t Thin-Film Interference Film between two different materials: t n = n1 Both waves experience /2 phase change at reflection. n = n2 n = n3 n3 > n2 > n1 Constructive interference: 2t m film m t m 2n2 n2 Destructive interference: 1 1 2t m film m 2 2 n2 1 m 2 t 2n2 Thin-Film Interference 1 Film between two different materials: Wave 1 experiences /2 phase change at reflection. Wave 2 does not. 2 t n = n1 n = n2 n = n3 n 2 > n3 > n 1 Constructive interference: 2t m film m 2 1 m 2 t 2n2 n2 Destructive interference: 1 1 m film m 2 2 2 n2 m t 2n2 2t Interferometers An interferometer separates a light wave into two parts and then recombines them, so that they interfere. How they interfere depends on what happened to them while they were separate. Many uses: measure wavelength of light measure displacements measure wavefront errors caused by optics under test fundamental experiments in physics Diffraction: Huygens’ Principle Consider a plane wavefront. Each point can be considered a source of spherical “wavelets” that move forward at the speed of the wave. The next wavefront is a plane tangent to the wavelets. Diffraction: Huygens’ Principle Christiaan Huygens 1629 – 1695 Dutch physicist and astronomer Early telescope maker First observed Saturn’s rings and one of Saturn’s moons Image from National Center for Atmospheric Research Diffraction: Huygens’ Principle For every direction except “straight ahead”: For each wavelet source point, we can find another one that will interfere destructively. That is, as long as the wavefront goes on forever … /2 Diffraction: Huygens’ Principle If something limits or truncates the wavefront, such as passing through an opening, points at and near the end will lack “partners in destruction” for some directions. The light spreads out: diffraction. Diffraction: Huygens’ Principle For each point in the upper half of the opening, in the direction , there is a point in the lower half that interferes destructively with it. A dark fringe occurs in this direction. Condition for dark fringes: sin m W Diffraction: Huygens’ Principle Resulting far-field diffraction pattern from a single slit: Diffraction: Circular Aperture If light passes through a circular, rather than rectangular, opening, a circular diffraction pattern results. The angular radius of the first dark ring: angle in radians 1.22 D aperture diameter Diffraction and Spatial Resolution What is the smallest angular distance between two objects, so that we can look at an image of them and tell that there are two instead of one? (Resolution) Diffraction and Spatial Resolution John William Strutt, Lord Rayleigh (1842 – 1919) English physicist Nobel Prize in physics, 1904 (gas theory and discovery of argon) Image from Nobel e-Museum Diffraction and Spatial Resolution Rayliegh’s Criterion for Resolution Two point objects are resolved if their angular separation is at least the radius of the first dark ring in their diffraction patterns. min 1.22 D Diffraction and Spatial Resolution What does this mean to someone who designs optical systems? Increase aperture size: improve diffraction performance (min gets smaller) Increase aperture size: geometric aberrations become worse (blur circle due to spherical, coma, etc.) While geometric blur is large compared to diffraction pattern, reducing aberrations strongly improves image quality When geometric blur has been made small compared to diffraction … diminishing returns for correction improvements. System is “diffraction-limited.” Diffractive Spectral Dispersion Diffraction grating: a large number of parallel slits or lines, each of which acts as a diffracting aperture. Condition for bright fringes: m sin d Diffractive Spectral Dispersion Condition for bright fringes: m sin d grating “pitch” Diffractive Spectral Dispersion Gratings can be either transmissive or reflective. They are used to separate light by wavelength (color) in optical instruments called spectrometers. Many natural processes can be identified and characterized from the discrete wavelengths of the light that they emit, or absorb, or both. The atomic structures of crystals can act as gratings and produce X-ray diffraction patterns. (Also useful in analytic science.)