Diffraction Light bends History of diffraction Diffraction solution

Diffraction Light bends! History of diffraction Diffraction solution to Maxwell's Equations Fraunhöfer Diffraction Some examples Young’s two-slit experiment Prof. Rick Trebino Georgia Tech www. frog. gatech. edu

Diffraction of Ocean-Water Waves Light does not always travel in a straight line. It tends to bend around objects. This tendency is called diffraction. Ocean waves passing through slits in Tel Aviv, Israel Diffraction occurs for all waves, whatever the phenomenon.

Diffraction of a Wave by a Slit Whether waves in water or electromagnetic radiation in air, passage through a slit yields a diffraction pattern that will appear more dramatic as the size of the slit approaches (decreases to) the wavelength of the wave.

Radio waves diffract around mountains. When the wavelength is a km long, a mountain peak is a very sharp edge! Another effect that occurs is scattering, so diffraction’s role is not obvious.

Even without a small slit, diffraction can be strong. Simple propagation past an edge yields an unintuitive intensityfringe pattern. Transmission Diffraction by an Edge t(x) x Laser light passing by edge Bending Fringes

Diffraction Shadow of a hand illuminated by a Helium -Neon laser. Note the fringes around the edges. Interesting fringes in a wave that passes near an edge:

Diffraction Gratings A diffraction grating is a series of equally spaced tiny slits.

Why It’s Hard to See Diffraction tends to cause ripples at edges. But a point source is required to see this effect. A large source masks them. Screen with hole Rays from a point source yield a perfect shadow of the hole. Rays from other regions blur the shadow. Example: a large source (like the sun) casts blurry shadows, masking the diffraction ripples

History of Diffraction of light was first characterized by Francesco Grimaldi, who also coined the term diffraction. Isaac Newton called it inflexion of light rays. James Gregory (1638– 1675) observed the diffraction patterns from a bird feather—the first diffraction grating to be discovered. Francesco Grimaldi 1618 -1663 Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits. He deduced that light must propagate as a wave. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction in 1815 -1818 giving great support to the wave theory of light advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

How to Profit from Diffraction… "Amazing x-ray glasses!” “See the bones in your hand, see through clothes!" Actual view through a bird feather Marketed in the 1950 s and 60 s by Harold von Braunhut, who also sold “Amazing Sea Monkeys. ”

Huygens’ Principle says that every point along a wave-front emits a spherical wave that interferes with all others. Christiaan Huygens 1629 – 1695 For a plane wave without an aperture, all these spherical waves still add up to a plane wave in the forward direction. Fresnel’s solution for diffraction from an aperture illustrates this idea.

Diffraction Geometry We wish to find the light electric field after a screen with a hole in it. This is a very general problem with far-reaching applications. Aperture transmission y y’ t(x, y) = 0 or 1 E(x, y) P Observation plane P’ x z E(x’, y’) Incident wave-fronts This region is assumed to be much smaller than this one. What is E(x’, y’) at a distance z from the plane of the aperture? x’

Diffraction Solution The field in the observation plane, E(x’, y’), at a distance z from the aperture plane is a sum of spherical waves from every point within the aperture: Spherical wave! where: How to simplify this result? Because z is much larger than all the other distances, we can approximate r by z in the denominator. But we can’t approximate r in the exp by z because it gets multiplied by k, which is big, so relatively small changes in r can make a big difference in the sines and cosines!

Diffraction Approximations But, inside the exp, we can write: This yields:

The Fraunhöfer Approximation Multiplying out the squares: Factoring out the quantities independent of x and y: If k (x 2 + y 2) / 2 z << 1, the quadratic terms << 1, so we can neglect them. This means going a distance away: z >> k (x 2 + y 2) /2 = p (x 2 + y 2) /l If (x 2 + y 2) < 1 mm 2 and l = 1 micron, then z >> 3 m.

Diffraction Conventions We’ll typically assume that a plane wave is incident on the aperture. There’s still an exp[i(k z –w t)], but this is constant with respect to x and y. And we’ll usually ignore the various factors in front (they don’t affect the shape of the diffracted intensity, that is, they don’t depend on x’ or y’):

The Fraunhöfer Diffraction Formula We can write this result in terms of the off-axis k-vector components: Aperture transmission function that is: and: kx kx = kx’/z and ky = ky’/z kz ky kx and ky are the off-axis k-vector components of the diffracted wave. Diffraction is simply a Fourier transform!

The Uncertainty Principle in Diffraction! kx = kx’/z Because the diffraction pattern is the Fourier transform of the slit, there’s an uncertainty principle between the slit width and diffraction pattern width! If the input field is a plane wave and Dx is the slit width and Dkx is the proportional to the beam angular width after the screen, Or: The smaller the slit, the larger the diffraction angle and the bigger the diffraction pattern!

Fraunhöfer Diffraction from a Slit Fraunhöfer Diffraction from a slit is simply the Fourier transform of a rect function, which is a sinc function. The intensity is then sinc 2. t(x) = rect(x/w) I(x’)

Fraunhöfer Diffraction from a Square Aperture The diffracted field is a sinc function in both x’ and y’ because the Fourier transform of a rect function is sinc. Diffracted field Diffracted intensity

Diffraction from a Circular Aperture A circular aperture yields a diffracted Airy Pattern, which looks a lot like a sinc function, but actually involves a Bessel function. Diffracted field Diffracted intensity

Diffraction from Small and Large Circular Far-field Apertures intensity pattern from a small aperture Recall the Scale Theorem! This is the Uncertainty Principle for diffraction. Far-field intensity pattern from a large aperture

Fraunhöfer Diffraction from Two Slits t(x) w w w 2 a -a 0 a x t(x) = rect[(x+a)/w] + rect[(x-a)/w] I(x’) kx’/z

One-Slit vs. Two-Slit Patterns kx kx

Diffraction from One and Two Slits Measured Fraunhöfer diffraction patterns One slit Two slits In 1803, Thomas Young measured the above two-slit pattern, which convincingly confirmed the wave nature of light (how could particles yield such a pattern? ), ending centuries of debate as to whether light was a particle or a wave.