Light Through a Single Slit Light passes through

Light Through a Single Slit • Light passes through a slit or opening and then illuminates a screen • As the width of the slit becomes closer to the wavelength of the light, the intensity pattern on the screen and additional maxima become noticeable Copyright © 2009 Pearson Education, Inc.

Single-Slit Diffraction • Water wave example of single-slit diffraction – All types of waves undergo single-slit diffraction – Water waves have a wavelength easily visible • Diffraction is the bending or spreading of a wave when it passes through an opening Copyright © 2009 Pearson Education, Inc.

Huygen’s Principle • It is useful to draw the wave fronts and rays for the incident and diffracting waves • Huygen’s Principle can be stated as all points on a wave front can be thought of as new sources of spherical waves Copyright © 2009 Pearson Education, Inc.

Double-Slit Interference • Light passes through two very narrow slits • When the two slits are both very narrow, each slit acts as a simple point source of new waves • The outgoing waves from each slit are like simple spherical waves • The double slit experiment showed conclusively that light is a wave • Experiment was first carried out by Thomas Young around 1800 Copyright © 2009 Pearson Education, Inc.

Young’s Double-Slit Experiment • Light is incident onto two slits and after passing through them strikes a screen • The light intensity on the screen shows an interference pattern Copyright © 2009 Pearson Education, Inc.

Young’s Experiment, cont. • The experiment satisfies the general requirements for interference – The interfering waves travel through different regions of space as they travel through different slits – The waves come together at a common point on the screen where they interfere – The waves are coherent because they come from the same source • Interference will determine how the intensity of light on the screen varies with position Copyright © 2009 Pearson Education, Inc.

Young’s Experiment • Assume the slits are very narrow • According the Huygen’s principle, each slit acts as a simple source with circular wave fronts as viewed from above • The light intensity on the screen alternates between bright and dark as you move along the screen – These areas correspond to regions of constructive interference and destructive interference Copyright © 2009 Pearson Education, Inc.

Double Slit Analysis • Need to determine the path length between each slit and the screen • Assume W is very large • If the slits are separated by a distance d, then the difference in length between the paths of the two rays is ΔL = d sin θ Copyright © 2009 Pearson Education, Inc.

Double Slit Analysis • If ΔL is equal to an integral number of complete wavelengths, then the waves will be in phase when they strike the screen – The interference will be constructive – The light intensity will be large • If ΔL is equal to a half number of complete wavelengths, then the waves will not be in phase when they strike the screen – The interference will be destructive – The light intensity will be zero Copyright © 2009 Pearson Education, Inc.

Conditions for Interference • For constructive interference, d sin θ = m λ – m = 0, ± 1, ± 2, … – Will observe a bright fringe • For destructive interference, d sin θ = (m + ½) λ – m = 0, ± 1, ± 2, … – Will observe a dark fringe Copyright © 2009 Pearson Education, Inc.

Double-Slit Intensity Pattern • The angle θ varies as you move along the screen • Each bright fringe corresponds to a different value of m • Negative values of m indicate that the path to those points on the screen from the lower slit is shorter than the path from the upper slit Copyright © 2009 Pearson Education, Inc.

Spacing Between Slits • Notation: – d is the distance between the slits – W is the distance between the slits and the screen – h is the distance between the adjacent bright fringes • For m = 1, • Since the angle is very small, sin θ ~ θ and θ ~ λ/d • Between m = 0 and m = 1, h = W tan θ Copyright © 2009 Pearson Education, Inc.

Approximations • Since the wavelength of light is small, the angles involved in the double-slit analysis are also small • For small angles, tan θ ~ θ and sin θ ~ θ • Using the approximations, h = W θ = W λ / d Copyright © 2009 Pearson Education, Inc.

Interference with Monochromatic Light • The conditions for interference state the interfering waves must have the same frequency – This means they must have the same wavelength • Light with a single frequency is called monochromatic (one color) • Light sources with a variety of wavelengths are generally not useful for double-slit interference experiments – The bright and dark fringes may overlap or the total pattern may be a “washed out” sum of bright and dark regions – No bright or dark fringes will be visible Copyright © 2009 Pearson Education, Inc.

Single-Slit Interference • Slits may be narrow enough to exhibit diffraction but not so narrow that they can be treated as a single point source of waves • Assume the single slit has a width, w • Light is diffracted as it passes through the slit and then propagates to the screen Copyright © 2009 Pearson Education, Inc.

Single-Slit Analysis • The key to the calculation of where the fringes occur is Huygen’s principle • All points across the slit act as wave sources • These different waves interfere at the screen • For analysis, divide the slit into two parts Copyright © 2009 Pearson Education, Inc.

Single-Slit Fringe Locations • If one point in each part of the slit satisfies the conditions for destructive interference, the waves from all similar sets of points will also interfere destructively • Destructive interference will produce a dark fringe Copyright © 2009 Pearson Education, Inc.

Single-Slit Analysis: Dark Fringers • Conditions for destructive interference are w sin θ = ±m λ – m = 1, 2, 3, … – The negative sign will correspond to a fringe below the center of the screen Copyright © 2009 Pearson Education, Inc.

Single-Slit Analysis: Bright Fringes • There is no simple formula for the angles at which the bright fringes occur • The intensity on the screen can be calculated by adding up all the Huygens waves • There is a central bright fringe with other bright fringes that are lower in intensity – The central fringe is called the central maximum – The central fringe is about 20 times more intense than the bright fringes on either side – The width of the central bright fringe is approximately the angular separation of the first dark fringes on either side Copyright © 2009 Pearson Education, Inc.

Single-Slit – Central Maximum • The full angular width of the central bright fringe = 2 λ / w • If the slit is much wider than the wavelength, the light beam essentially passes straight through the slit with almost no effect from diffraction Copyright © 2009 Pearson Education, Inc.

Double-Slit Interference with Wide Slits • When the slits of a double-slit experiment are not extremely narrow, the single-slit diffraction pattern produced by each sit is combined with the sinusoidal double-slit interference pattern • A full calculation of the intensity pattern is very complicated Copyright © 2009 Pearson Education, Inc.

Diffraction Grating • An arrangement of many slits is called a diffraction grating • Assumptions – The slits are narrow • Each one produces a single outgoing wave – The screen is very far away Copyright © 2009 Pearson Education, Inc.

Diffraction Grating • Since the screen is far away, the rays striking the screen are approximately parallel – All make an angle θ with the horizontal axis • If the slit-to-slit spacing is d, then the path length difference for the rays from two adjacent slits is ΔL = d sin θ • If ΔL is equal to an integral number of wavelengths, constructive interference occurs • For a bright fringe, d sin θ = m λ – m = 0, ± 1, ± 2, … Copyright © 2009 Pearson Education, Inc.

Diffraction Grating, final • The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit • The overall intensity pattern depends on the number of slits • The larger the number of slits, the narrower the peaks Copyright © 2009 Pearson Education, Inc.

Grating and Color Separation • A diffraction grating will produce an intensity pattern on the screen for each color • The different colors will have different angles and different places on the screen • Diffraction gratings are widely used to analyze the colors in a beam of light Copyright © 2009 Pearson Education, Inc.

Diffraction and CDs • Light reflected from the arcs in a CD acts as sources of Huygens waves • The reflected waves exhibit constructive interference at certain angles • Light reflected from a CD has the colors “separated” Copyright © 2009 Pearson Education, Inc.

Crystal Diffraction of X-rays • Diffraction effects occur with other types of waves • The atoms of a crystal are arranged in a periodic way, forming planes • These planes reflect em radiation • Leads to interference of the reflected rays Copyright © 2009 Pearson Education, Inc.

X-Ray Diffraction, cont. • The effective slit spacing is the distance between atomic planes – Typically 3 x 10 -10 m – Compared to 10 -4 m or 10 -5 m for a grating • X-rays have the appropriate wavelength to diffract – The planes give dots instead of fringes • By measuring the angles that give constructive interference, the distance between the planes can be measured Copyright © 2009 Pearson Education, Inc.

Optical Resolution • For a circular opening of diameter D, the angle between the central bright maximum and the first minimum is – The circular geometry leads to the additional numerical factor of 1. 22 Copyright © 2009 Pearson Education, Inc.

Telescope Example • • • Assume you are looking at a star through a telescope Diffraction at the opening produces a circular diffraction spot Assume there actually two stars The two waves are incoherent and do not interfere Each source produces its own different pattern Copyright © 2009 Pearson Education, Inc.

Rayleigh Criterion • If the two sources are sufficiently far apart, they can be seen as two separate diffraction spots (A) • If the sources are too close together, their diffraction spots will overlap so much that they appear as a single spot (C) Copyright © 2009 Pearson Education, Inc.

Rayleigh Criterion, cont. • Two sources will be resolved as two distinct sources of light if their angular separation is greater than the angular spread of a single diffraction spot • This result is called the Rayleigh criterion • For a circular opening, the Rayleigh criterion for the angular resolution is • Two objects will be resolved when viewed through an opening of diameter D if the light rays from the two objects are separated by an angle at least as large as θmin Copyright © 2009 Pearson Education, Inc.

Limits on Focusing • A perfect lens will focus a narrow parallel beam of light to a precise point at the focal point of the lens • The ray optics approximation ignores diffraction • The real focus is spread over a disc Copyright © 2009 Pearson Education, Inc.

Limits on Focusing, cont. • If the lens has a diameter D, it acts like an opening and according to the Rayleigh criterion produces a diffracted beam spread over a range of angles • Diffraction spreads the focal point over a disk of radius r • • The focal length is limited to Copyright © 2009 Pearson Education, Inc.

Limits on Focusing, final • The wave nature of light limits the focusing qualities of even a perfect lens • It is not possible to focus a beam of light to a spot smaller than approximately the wavelength • The ray approximation of geometrical optics can be applied at size scale much greater than the wavelength • When a slit or a focused beam of light is made so small that its dimensions are comparable to the wavelength, diffraction effects become important Copyright © 2009 Pearson Education, Inc.

Scattering • When the wavelength is larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves • The amplitude of the scattered wave depends on the size of the scattering object compared to the wavelength • Blue light is scattered more than red – Called Rayleigh scattering Copyright © 2009 Pearson Education, Inc.

Blue Sky • The light we see from the sky is sunlight scattered by the molecules in the atmosphere • The molecules are much smaller than the wavelength of visible light – They scatter blue light more strongly than red – This gives the atmosphere its blue color Copyright © 2009 Pearson Education, Inc.

Scattering, Sky, and Sun • Blue sky – Although violet scatters more than blue, the sky appears blue – The Sun emits more strongly in blue than violet – Our eyes are more sensitive to blue – The sky appears blue even though the violet light is scattered more • Sun near horizon – There are molecules to scatter the light – Most of the blue is scattered away, leaving the red Copyright © 2009 Pearson Education, Inc.

Nature of Light • Interference and diffraction show convincingly that light has wave properties • Certain properties of light can only be explained with a particle theory of light – Color vision is one effect that can be correctly explained by the particle theory • Have strong evidence that light is both a particle and a wave – Called wave-particle duality – Quantum theory tries to reconcile these ideas Copyright © 2009 Pearson Education, Inc.

Color Vision • Color vision is due to light detectors in the eye called cones • The three types of cones are sensitive to light from different regions of the visible spectrum • Particles of light, photons, carry energy that depends on the frequency of the light Copyright © 2009 Pearson Education, Inc.