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

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

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

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

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

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

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

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

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

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 Will observe a dark fringe

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

Spacing Between Slits Notation: For m = 1, 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

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

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

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

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

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

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

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

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

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

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

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, …

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

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

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”

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

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

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

Telescope Example Assume you are looking at a star through a telescope Diffraction at the opening produces a circular diffraction spot Assume there are actually two stars The two waves are incoherent and do not interfere Each source produces its own different pattern

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)

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

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

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

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

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

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

Scattering, Sky, and Sun Blue sky Sun near horizon 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 more molecules to scatter the light Most of the blue is scattered away, leaving the red

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

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