| The shimmering iridescent colors of peacock feathers are difficult to reproduce exactly in paint or in photographs, because they are not due to pigmentation, but to selective reflection due to interference in thin oil films. |
The upper air-water surface in the diagram at right is a closed boundary. Light waves are reflected with a phase change of 180°. The lower water-air surface is an open boundary. Light waves are returned without a phase change. If the path difference between reflections 1 and 2 in the diagram at right is one half wavelength, the two reflections will be in phase, and maximum light intensity will be reflected.
Zero path difference:
The path difference is zero, for a film of zero thickness. The two reflections, 1 and 2, will be out of phase and will cancel. No light is reflected.
Note: reflections (1 and 2) are not of exactly the same amplitude and will not cancel completely. It can be shown that the sum of the amplitudes 2+3+4+ etc. is exactly equal to the amplitude of reflection 1 when the film is of zero thickness and the phase of reflections 2, 3, 4 etc. are all the same. A perfectly transparent film of 'zero' thickness can be shown to reflect no light, as expected.
Half l path difference
If the film thickness is an uneven multiple of one quarter wavelength (three, five, seven quarters... etc.) the path difference will be a multiple of one half wavelength and the two reflections, 1 and 2, will be out of phase; they will cancel.
As a soap film hangs vertically it becomes very thin at the top and increases in thickness towards the bottom. The top half of the film reflects very little light and appears dark. The lower half of the film is crossed by a series of colored bands.
As the film ages, the colored bands move down and are replaced by a non reflecting upper region. If the film is thick (ten wavelengths of red light) there are many wavelengths in the visible range for which maximum reflection takes place and few colors are seen.
Note: the coherence length is about 0.1 mm for sunlight. A glass plate with a thickness of 1 mm is a 'thin' film if we are using monochromatic laser light with a coherence length of a cm or more. Interference fringes are sometimes seen when reflecting coherent laser light from the two surfaces of window glass!
A thin transparent film has formed on ceramic tiles set into a table that has been left outside for five years in Bangkok air. Similar films on car windows are often referred to as 'road film'.
Vivid colors are seen when the sky is reflected in the tiles. The colors shift and change as the point of view changes. A detail shows that the interference fringes are formed on the surface (in sharp focus). The reflected image of an overhead palm frond (below) is in focus but the interference fringes are not.
Oil on water: An oil slick has formed on water left for a week in a brass bowl. The thickness of the oil determines the color in reflection.
Glass beads: the colors are due to interference in thin films, of different thickness, and also to the different viewing angles.
Anodized titanium: the titanium keys have a thin, uniform, transparent titanium oxide layer on the surface. The colors in reflection are due to selective interference.
Exhaust pipe: the custom made exhaust pipe is colored by interference in a thin oxide film of varying thickness.
Paua shell: the polished shell of the NZ Paua [known as abalone in the US] is colored in reflection by selective interference in thin layers of calcium carbonate.
Feathers and butterfly wings: butterfly wings and peacock feathers are colored by interference in thin oil films on the wings and feathers.
Beetle carapices: These three beetles appear to be the same species but the colors by interference are diffetent due to the different thicknes of the thin films.
About 4% of perpendicular incident light is returned from each surface of a glass lens (n =1.50) in air. Coating a lens with a layer of magnesium fluoride (n =1.32) one quarter wavelength thick, suppresses the reflection since both upper and lower MgF2 interfaces are closed boundaries. Zero reflection is possible for only one visible wavelength. Reflections from bloomed lenses are colored.
 Compound camera lenses reflect images of an overhead light in many colors, depending on the exact thickness to the MgF2 layers which are not the same for each surface. |
A coated lens is said to be 'bloomed' because the appearance of the surface in reflected light is similar to the bloom on ripe fruit (plums, grapes, etc) due to an oil layer on the skin.
Mirror sunglasses and the one-way mirrors found in Police stations are made with multiple interference layers. It is possible to make such a mirror 100% reflecting at one wavelength. The terminal mirrors of a gas laser cavity are made this way; carefully constructed to reflect at close to 100% at the laser wavelength. The mirrors are semitransparent at other wavelengths.
The one-way mirrors on common brands of sunglasses are made with multiple thin films. The glasses shown reflect predominantly in the red at normal incidence. The reflection fades through orange to yellow at around 45°. Clearly, since the wavelength of yellow light is shorter than red the path difference at higher angles of incidence is less than at normal incidence. The point is illustrated with two diagrams.
Paua [abalone] shell shows the same effect. The path difference for vertical incidence in sunlight returns predominantly pink from this shell fragment. Mostly green light is reflected from the same shell at angles of incidence of 70 - 80°.
Exercise
A lens coating is a single layer, one half wavelength thick at normal incidence for red light (l = 640 nm), the effective path difference at 45° is clearly less.
(i) Estimate the path difference at normal incidence and at 45°.
(ii) If the refractive index of the thin film is 1.32, over glass, refractive index 1.5, find the physical thickness of the film.
(iii) Show that the relationship between the path difference D for the first two reflections and the angle of refraction, r, is given by....
The air gap between an optical flat and a convex lens is a thin film of air. Interference is most easily seen in reflection over a very small central area.
For a lens of one meter focal length the visible pattern in white light has a diameter of 1-2 mm. The fringes are easy to see on close inspection if it is realized that they are formed in the air gap, not at infinity or somewhere else. To see them, look closely and focus your eyes on the point of contact.
The photograph was taken in reflected white light using slide film. A low power microscope replaced the camera lens. Note the colors and the rapid fading of the pattern. Using a green filter sharpens the pattern somewhat.
Exercise
Show that for Newton's rings in the air gap between a spherical lens and a flat plate, the number of the fringe, counting from zero in the center, (which is dark) is proportional to the radius squared, both theoretically, and by taking measurements on the green photograph.
Air gaps
Irregular fringes between two microscope slides form a contour map of the air gap. The lighting in the unusual image (below left), gives the impression of the contour (hill). Photographs of the fringes usually appear to be flat (below right).
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Air wedges
If two optical flats (flat to within a quarter of a wavelength) are placed with a thin object at one end to form an air wedge the fringes are straight an parallel, separated by half a wavelength. Counting the fringes gives a measure to the thickness of the object. In this way the thickness of tissue paper can be measured more accurately than with a micrometer. The most convenient light source for measurements of this kind is a yellow sodium vapor lamp.
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A word to the wise The air wedge is a favorite exam question. Try these questions from IB papers. |