Refraction And Diffraction

A simple ripple tank

Diffraction of sound waves and of light waves will be discussed in a later unit of The Physics Classroom Tutorial. Reflection, refraction and diffraction are all boundary behaviors of waves associated with the bending of the path of a wave. The bending of the path is an observable behavior when the medium is a two- or three-dimensional medium. Reflection, Refraction, and Diffraction Like any wave, a sound wave doesn't just stop when it reaches the end of the medium or when it encounters an obstacle in its path. Rather, a sound wave will undergo certain behaviors when it encounters the end of the medium or an obstacle. This paper treats the derivation of a two-dimensional differential equation, which describes the phenomenon of combined refraction — diffraction for simple harmonic waves, and a method of solving this equation. Refraction and diffraction Book page 109 – 110 Syllabus 3.17, 3.21 ©cgrahamphysics.com 2016.

In physics, a ripple tank is a shallow glass tank of water used to demonstrate the basic properties of waves. It is a specialized form of a wave tank. The ripple tank is usually illuminated from above, so that the light shines through the water. Some small ripple tanks fit onto the top of an overhead projector, i.e. they are illuminated from below. The ripples on the water show up as shadows on the screen underneath the tank. All the basic properties of waves, including reflection, refraction, interference and diffraction, can be demonstrated.

Ripples may be generated by a piece of wood that is suspended above the tank on elastic bands so that it is just touching the surface. Screwed to wood is a motor that has an off centre weight attached to the axle. As the axle rotates the motor wobbles, shaking the wood and generating ripples.

Demonstrating wave properties

A number of wave properties can be demonstrated with a ripple tank. These include plane waves, reflection, refraction, interference and diffraction.

Plane waves

Image of plane waves

When the rippler is lowered so that it just touches the surface of the water, plane waves will be produced.

• Close-up of the rippler – the brown rectangle is an oscillating paddle

Circular waves

Ripple tank with a spherical source producing circular waves

When the rippler is attached with a point spherical ball and lowered so that it just touches the surface of the water, circular waves will be produced.

Reflection

Waves that are short relative to the size of an object will wrap around the object.
Waves that are short in respect to the size of an object will cast a shadow behind the object.
Numerical approximation of diffraction pattern from a slit of width equal to five times the wavelength of an incident plane wave in 3D visualization
Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D visualization

Demonstrating reflection and focusing of mirrors

By placing a metal bar in the tank and tapping the wooden bar a pulse of three of four ripples can be sent towards the metal bar. The ripples reflect from the bar. If the bar is placed at an angle to the wavefront the reflected waves can be seen to obey the law of reflection. The angle of incidence and angle of reflection will be the same.

If a concave parabolic obstacle is used, a plane wave pulse will converge on a point after reflection. This point is the focal point of the mirror. Circular waves can be produced by dropping a single drop of water into the ripple tank. If this is done at the focal point of the 'mirror' plane waves will be reflected back.

Refraction

If a sheet of glass is placed in the tank, the depth of water in the tank will be shallower over the glass than elsewhere. The speed of a wave in water depends on the depth, so the ripples slow down as they pass over the glass. This causes the wavelength to decrease. If the junction between the deep and shallow water is at an angle to the wavefront, the waves will refract. In the diagram above, the waves can be seen to bend towards the normal. The normal is shown as a dotted line. The dashed line is the direction that the waves would travel if they had not met the angled piece of glass.

In practice, showing refraction with a ripple tank is quite tricky to do.

• The sheet of glass needs to be quite thick, with the water over it as shallow as possible. This maximizes the depth difference and so causes a greater velocity difference and therefore greater angle.
• If the water is too shallow, viscous drag effects cause the ripples to disappear very quickly.
• The glass should have smooth edges to minimize reflections at the edge. Refraction And Diffraction Similarities

If a small obstacle is placed in the path of the ripples, and a slow frequency is used, there is no shadow area as the ripples refract around it, as shown below on the right. A faster frequency may result in a shadow, as shown below on the right. If a large obstacle is placed in the tank, a shadow area will probably be observed.

If an obstacle with a small gap is placed in the tank the ripples emerge in an almost semicircular pattern. If the gap is large however, the diffraction is much more limited. Small, in this context, means that the size of the obstacle is comparable to the wavelength of the ripples.

Diffraction from a grid

A phenomenon identical to the x-ray diffraction of x-rays from an atomic crystal lattice can also be seen, thus demonstrating the principles of crystallography. If one lowers a grid of obstacles into the water, with the spacing between the obstacles roughly corresponding to the wavelength of the water waves, one will see diffraction from the grid. At certain angles between the grid and the oncoming waves, the waves will appear to reflect off the grid; at other angles, the waves will pass through. Similarly, if the frequency (wavelength) of the waves is altered, the waves will also alternately pass through or be reflected, depending on the precise relationship between spacing, orientation and wavelength.

Interference

Interference can be produced by the use of two dippers that are attached to the main ripple bar. In the diagrams below on the left the light areas represent crests of waves, the black areas represent troughs. Notice the grey areas: they are areas of destructive interference where the waves from the two sources cancel one another out. To the right is a photograph of two-point interference generated in a circular ripple tank.

• PSSC Physics – pioneering secondary-school physics textbook which made extensive use of ripple tanks to illustrate waves

References

• Breithaupt, Jim (2000) New Understanding Physics for Advanced Level pages 309–312, Nelson Thornes. ISBN9780748743148.

We classically think of light as always traveling in straight lines, but when light waves pass near a barrier they tend to bend around that barrier and become spread out. Diffraction of light occurs when a light wave passes by a corner or through an opening or slit that is physically the approximate size of, or even smaller than that light's wavelength.

A very simple demonstration of diffraction can be conducted by holding your hand in front of a light source and slowly closing two fingers while observing the light transmitted between them. As the fingers approach each other and come very close together, you begin to see a series of dark lines parallel to the fingers. The parallel lines are actually diffraction patterns. This phenomenon can also occur when light is 'bent' around particles that are on the same order of magnitude as the wavelength of the light. A good example of this is the diffraction of sunlight by clouds that we often refer to as a silver lining, illustrated in Figure 1 with a beautiful sunset over the ocean.

We can often observe pastel shades of blue, pink, purple, and green in clouds that are generated when light is diffracted from water droplets in the clouds. The amount of diffraction depends on the wavelength of light, with shorter wavelengths being diffracted at a greater angle than longer ones (in effect, blue and violet light are diffracted at a higher angle than is red light). As a light wave traveling through the atmosphere encounters a droplet of water, as illustrated below, it is first refracted at the water:air interface, then it is reflected as it again encounters the interface. The beam, still traveling inside the water droplet, is once again refracted as it strikes the interface for a third time. This last interaction with the interface refracts the light back into the atmosphere, but it also diffracts a portion of the light as illustrated below. This diffraction element leads to a phenomenon known as Cellini's halo (also known as the Heiligenschein effect) where a bright ring of light surrounds the shadow of the observer's head.

The terms diffraction and scattering are often used interchangeably and are considered to be almost synonymous. Diffraction describes a specialized case of light scattering in which an object with regularly repeating features (such as a diffraction grating) produces an orderly diffraction of light in a diffraction pattern. In the real world most objects are very complex in shape and should be considered to be composed of many individual diffraction features that can collectively produce a random scattering of light.

One of the classic and most fundamental concepts involving diffraction is the single-slit optical diffraction experiment, first conducted in the early nineteenth century. When a light wave propagates through a slit (or aperture) the result depends upon the physical size of the aperture with respect to the wavelength of the incident beam. This is illustrated in Figure 3 assuming a coherent, monochromatic wave emitted from point source S, similar to light that would be produced by a laser, passes through aperture d and is diffracted, with the primary incident light beam landing at point P and the first secondary maxima occurring at point Q.

As shown in the left side of the figure, when the wavelength (λ) is much smaller than the aperture width (d), the wave simply travels onward in a straight line, just as it would if it were a particle or no aperture were present. However, when the wavelength exceeds the size of the aperture, we experience diffraction of the light according to the equation:

sinθ = λ/d

Where θ is the angle between the incident central propagation direction and the first minimum of the diffraction pattern. The experiment produces a bright central maximum which is flanked on both sides by secondary maxima, with the intensity of each succeeding secondary maximum decreasing as the distance from the center increases. Figure 4 illustrates this point with a plot of beam intensity versus diffraction radius. Note that the minima occurring between secondary maxima are located in multiples of π.

This experiment was first explained by Augustin Fresnel who, along with Thomas Young, produced important evidence confirming that light travels in waves. From the figures above, we see how a coherent, monochromatic light (in this example, laser illumination) emitted from point L is diffracted by aperture d. Fresnel assumed that the amplitude of the first order maxima at point Q (defined as εQ) would be given by the equation:

Refraction And Diffraction Examples

dεQ = α(A/r)f(χ)d

where A is the amplitude of the incident wave, r is the distance between d and Q, and f(χ) is a function of χ, an inclination factor introduced by Fresnel.

Diffraction of Light

Explore how a beam of light is diffracted when it passes through a narrow slit or aperture. Adjust the wavelength and aperture size and observe how this affects the diffraction intensity pattern.

Diffraction of light plays a paramount role in limiting the resolving power of any optical instrument (for example: cameras, binoculars, telescopes, microscopes, and the eye). The resolving power is the optical instrument's ability to produce separate images of two adjacent points. This is often determined by the quality of the lenses and mirrors in the instrument as well as the properties of the surrounding medium (usually air). The wave-like nature of light forces an ultimate limit to the resolving power of all optical instruments.

Our discussions of diffraction have used a slit as the aperture through which light is diffracted. However, all optical instruments have circular apertures, for example the pupil of an eye or the circular diaphragm and lenses of a microscope. Circular apertures produce diffraction patterns similar to those described above, except the pattern naturally exhibits a circular symmetry. Mathematical analysis of the diffraction patterns produced by a circular aperture is described by the equation:

sinθ(1) = 1.22(λ/d) where θ(1) is the angular position of the first order diffraction minima (the first dark ring), λ is the wavelength of the incident light, d is the diameter of the aperture, and 1.22 is a constant. Under most circumstances, the angle θ(1) is very small so the approximation that the sin and tan of the angle are almost equal yields:

From these equations it becomes apparent that the central maximum is directly proportional to λ/d making this maximum more spread out for longer wavelengths and for smaller apertures. The secondary mimina of diffraction set a limit to the useful magnification of objective lenses in optical microscopy, due to inherent diffraction of light by these lenses. No matter how perfect the lens may be, the image of a point source of light produced by the lens is accompanied by secondary and higher order maxima. This could be eliminated only if the lens had an infinite diameter. Two objects separated by a distance less than θ(1) can not be resolved, no matter how high the power of magnification. While these equations were derived for the image of a point source of light an infinite distance from the aperture, it is a reasonable approximation of the resolving power of a microscope when d is substituted for the diameter of the objective lens.

Thus, if two objects reside a distance D apart from each other and are at a distance L from an observer, the angle (expressed in radians) between them is:

θ = D / L

which leads us to be able to condense the last two equations to yield:

Where D(0) is the minimum separation distance between the objects that will allow them to be resolved. Using this equation, the human eye can resolve objects separated by a distance of 0.056 millimeters, however the photoreceptors in the retina are not quite close enough together to permit this degree of resolution, and 0.1 millimeters is a more realistic number under normal circumstances.

The resolving power of optical microscopes is determined by a number of factors including those discussed, but in the most ideal circumstances, this number is about 0.2 micrometers. This number must take into account optical alignment of the microscope, quality of the lenses, as well as the predominant wavelengths of light used to image the specimen. While it is often not necessary to calculate the exact resolving power of each objective (and would be a waste of time in most instances), it is important to understand the capabilities of the microscope lenses as they apply to the real world.

Contributing Authors

Mortimer Abramowitz - Olympus America, Inc., Two Corporate Center Drive., Melville, New York, 11747.

Refraction And Diffraction Chart

Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.