Diffraction Chapter 38 Huygens construction may be used to find the wave observed on the downstream side of an aperture of any shape. Diffraction The interference pattern encodes the shape as a Fourier transform essentially. pt In the plane of a slit for example, the wave amplitude has a box shape distribution so has a range of transverse wave vectors (momenta components) inversely proportional to the slit size. The transverse components of the momenta lead to a spreading of the wave. May 3, 2012 2 Single-slit diffraction Diffraction by a circular aperture In the forward direction, the intensity is maximal. The diffraction pattern of a circular aperture is crudely a rotated single slit diffraction pattern. To find the position of the first minimum, consider two sources a half slit width apart and find the angle for destructive interference. A refracting or reflecting optical element has an aperture that creates a diffraction pattern at the image plane of a distant point source. The overlap of these patterns determines the minimum resolvable angular separation between two point sources - the diffraction limit -which is roughly the ratio of wavelength to aperture size. All such pairs of sources interfere destructively and so the sum from all sources taken in pairs exhibits a minimum at that angle. 3 4
Diffraction from various apertures Double-slit diffraction Two narrow slits produce a simple interference pattern. It results from the interference of two cylindrical waves. Here are diffraction patterns resulting from various apertures The interference pattern created by two finite slits results from the superposition of the two diffracted waves. 5 Two frequencies appear - one associated with the slit separation and another with the slit width. 6 Gratings Diffraction Grating, Types An array of apertures such as slits produces a superposition of waves, one from each aperture, which encode the aperture distribution. A regular array produces constructive interference only for special angles. High finesse (sharp) features appear in the intensity distribution. The more apertures, the finer the features. 7 A transmission grating can be made by cutting parallel grooves on a glass plate.! The spaces between the grooves are transparent to the light and so act as separate slits. A reflection grating can be made by cutting parallel grooves on the surface of a reflective material.! The spaces between the grooves act as parallel sources of reflected light, like the slits in a transmission grating. Section 38.4
Diffraction Grating, cont. Diffraction Grating Spectrometer The collimated beam is incident on the grating. The condition for maxima is The diffracted light leaves the gratings and the telescope is used to view the image.! d sin θbright = mλ! m = 0, ±1, ±2, The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders. The integer m is the order number of the diffraction pattern. If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle. A high finesse grating enables an extremely precise measurement of wavelength, important in identifying and distinguishing the features of Section atomic and molecular spectra. Section 38.4 X-ray diffraction 38.4 Polarization of Light Waves The direction of polarization of each individual wave is defined to be the direction in which the electric field is vibrating. Scattering from a 3-D lattice of objects such as atoms in a crystal produces a high finesse interference pattern. In this example, the direction of polarization is along the y-axis. The locations of maxima may be found by considering planes of sources. All individual electromagnetic waves traveling in the x direction have an electric field vector parallel to the yz plane. Bragg s Law 11 This vector could be at any possible angle with respect to the y axis. Section 38.6
Unpolarized Light, Example Polarization of Light, cont. All directions of vibration from a wave source are possible. The resultant em wave is a superposition of waves vibrating in many different directions. This is an unpolarized wave. The arrows show a few possible directions of the waves in the beam. A wave is said to be linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point. The plane formed by the field and the direction of propagation is called the plane of polarization of the wave. Methods of Polarization Polarization by Selective Absorption It is possible to obtain a linearly polarized beam from an unpolarized beam by removing all waves from the beam except those whose electric field vectors oscillate in a single plane. Processes for accomplishing this include:! Selective absorption! Reflection! Double refraction! Scattering The most common technique for polarizing light. Uses a material that transmits waves whose electric field vectors lie in the plane parallel to a certain direction and absorbs waves whose electric field vectors are in all other directions.
Selective Absorption, cont. Intensity of a Polarized Beam E. H. Land discovered a material that polarizes light through selective absorption.! He called the material Polaroid.! The molecules readily absorb light whose electric field vector is parallel to their lengths and allow light through whose electric field vector is perpendicular to their lengths. It is common to refer to the direction perpendicular to the molecular chains as the transmission axis. In an ideal polarizer,! All light with the electric field parallel to the transmission axis is transmitted.! All light with the electric field perpendicular to the transmission axis is absorbed. The intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as! I = I max cos 2 θ! I max is the intensity of the polarized wave incident on the analyzer.! This is known as Malus law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other. The intensity of the transmitted beam is a maximum when the transmission axes are parallel.! θ = 0 or 180 o The intensity is zero when the transmission axes are perpendicular to each other.! This would cause complete absorption. Intensity of Polarized Light, Examples Polarization by Reflection When an unpolarized light beam is reflected from a surface, the reflected light may be! Completely polarized! Partially polarized! Unpolarized The polarization depends on the angle of incidence. On the left, the transmission axes are aligned and maximum intensity occurs. In the middle, the axes are at 45 o to each other and less intensity occurs. On the right, the transmission axes are perpendicular and the light intensity is a minimum.! If the angle is 0, the reflected beam is unpolarized.! For other angles, there is some degree of polarization.! For one particular angle, the beam is completely polarized.
Polarization by Reflection, cont. Polarization by Reflection, Partially Polarized Example The angle of incidence for which the reflected beam is completely polarized is called the polarizing angle, θ p. =atan (n2/n1). Brewster s law relates the polarizing angle to the index of refraction for the material. θ p may also be called Brewster s angle. Unpolarized light is incident on a reflecting surface. The reflected beam is partially polarized. The refracted beam is partially polarized. Polarized sunglasses are designed to absorb the dominant reflected polarization and thereby reduce glare. Polarization by Reflection, Completely Polarized Example Polarization by Double Refraction Unpolarized light is incident on a reflecting surface. The reflected beam is completely polarized. The refracted beam is perpendicular to the reflected beam. The angle of incidence is Brewster s angle. In certain crystalline structures, the speed of light is not the same in all directions. Such materials are characterized by two indices of refraction. They are often called double-refracting or birefringent materials.
Polarization by Double Refraction, cont. Polarization by Double Refraction, Rays The ordinary (O) ray is characterized by an index of refraction of n o. Unpolarized light splits into two plane-polarized rays. The two rays are in mutual perpendicular directions.! This is the same in all directions. The second ray is the extraordinary (E) ray which travels at different speeds in different directions.! Characterized by an index of refraction of n E that varies with the direction of propagation. Polarization by Double Refraction, Optic Axis Some Indices of Refraction There is one direction, called the optic axis, along which the ordinary and extraordinary rays have the same speed.! n O = n E The difference in speeds for the two rays is a maximum in the direction perpendicular to the optic axis.
Optical Stress Analysis Polarization by Scattering Some materials become birefringent when stressed. When a material is stressed, a series of light and dark bands is observed. When light is incident on any material, the electrons in the material can absorb and reradiate part of the light.! This process is called scattering.! The light bands correspond to areas of greatest stress. Optical stress analysis uses plastic models to test for regions of potential weaknesses. Polarization by Scattering, cont. The horizontal part of the electric field vector in the incident wave causes the charges to vibrate horizontally. The vertical part of the vector simultaneously causes them to vibrate vertically. If the observer looks straight up, he sees light that is completely polarized in the horizontal direction. An example of scattering is the sunlight reaching an observer on the Earth being partially polarized. Scattering, cont. Short wavelengths (violet) are scattered more efficiently than long wavelengths (red). When sunlight is scattered by gas molecules in the air, the violet is scattered more intensely than the red. When you look up, you see blue.! Your eyes are more sensitive to blue, so you see blue instead of violet. At sunrise or sunset, much of the blue is scattered away, leaving the light at the red end of the spectrum.
Spherical aberration Optical Activity Several geometrical effects lead to blur in imaging systems. Certain materials display the property of optical activity.! A material is said to be optically active if it rotates the plane of polarization of any light transmitted through it.! Molecular asymmetry determines whether a material is optically active. A spherical mirror (or refracting lens) focuses light from the perimeter closer - the focal length is a function of radius leading to blur called spherical aberration. 34 Schmidt corrector Chromatic aberration Corrected Because refractive index varies with frequency, different colors Uncorrected from the same source are focused at different places. A lens can be used to adjust the wavefront phase before reflection by a spherical surface to correct for spherical aberration. An achromatic doublet is a pair of lens with compensating chromaticity. A cost is the introduction of a chromatic aberration. Chromatic aberration of the human eye is close to the sensor segmentation limit. 35 36
Coma Astigmatism Off axis astigmatic defocus is associated with the geometric projection of wavefronts onto a reflecting or refracting surface and onto an image plane. Wavefronts from off-axis object point are imaged not to a point but blurred proportional to angle. Consequently, the image of an object develops a comet-like tail. A cylindrical imperfection in the spherical reflecting or refracting system produces a similar effect. The focus for light along one axis is different from that for light along an orthogonal axis. A lens doublet can compensate for both spherical aberration and coma. 37 Aberration classification Vertical lines focused, horizontal lines defocused 38 Matrix mechanics An optical element performs a transformation of phase space (position r and angle r ). Aberrations may be classified as terms in the expansion of the deviation of the wavefront surface from ideal. A refracting surface or drift space may be represented by a matrix. Combinations of simple spherical surface optic elements may be used to compensate order by order. The transformation due to a collection is represented by the matrix product. Only forward going light is considered. 39 40