PHYSICS AND THE VISUAL ARTS
Notes on LESSON 6
Things to remember from lecture and demonstrations in Lesson 6.
We saw earlier that light is a traveling wave composed of an oscillating
electric field (and a magnetic field as well). Two light waves derived from
the same source are said to be coherent because the have exactly the same
wavelength and maintain a constant phase relative to each other. When two
coherent waves occur at the same place and same time, they interfere. That
is, the resulting amplitude is the algebraic sum of their individual amplitudes.
Constructive interference occurs when the waves are exactly in phase so that their
amplitudes add. Destructive interference occurs when the waves are exactly out of phase (by
one half wavelength) and their amplitudes subtract.
You saw the interference of laser light after it passed
through two narrow slits. When the light passed through only a single
slit the otherwise narrow beam spread into a wide beam perpendicular to
the slit length, i.e. a narrow vertical slit spreads the light into a
wide horizontal beam. This behavior of the light is called
diffraction. When two close-by slits are illuminated with the same
laser beam, the two diffractively-spread beams interfere at the screen to
give a pattern of dots. This phenomenon is known as double-slit
interference. See Figure 5.2 on p. 107 of your text for a example of
a double slit interference pattern. For a beam aimed normally toward a screen a distance L
away (see figure below), the positions of dots on the screen measured from the central dot
of the undiffracted beam may be obtained from the equation:
y = (nλ)(L/d)
where n is an integer referring to the nth dot, d is the separation
between the slits, and λ (lambda) is the wavelength.
The distance between adjacent spots corresponds to a change from n to
n+1. It is a distance equal to
dot spacing = (λ)(L/d).
The dot spacing is also referred to as the fringe spacing.
The interference arising from the double slit is often called Young's double slit pattern after Thomas Young
who published a description of his experiments with a double slit in 1807. It was Young who used the
double-slit interference to measure the wavelength of light for the very first time.
Remember, the spacing in the interference pattern depends on wavelength of the light and the spacing between
the slits. From the equations above, we see that the spacing of the interference pattern depend directly on
the wavelength and on the distance between the slits and the screen and inversely with the separation between
the slits.
One recognizable feature of laser light is the "speckle pattern" seen when the light is spread out and
reflected from a smooth surface. This pattern is formed in your eye (or as seen in the video, in the camera)
as the coherent laser light interferes with itself as it is reflected from different positions and thus
travels different path lengths to your eye. You don't see this speckle with ordinary incoherent lights. When
you observe the speckle pattern you know that you are seeing coherent light, that is, you are seeing laser
light.
Thin film colors of soap bubbles, oil slicks, and
interference filters arise from the interference of light reflected off
the first surface and off the second surface of the thin films.(See Plate 5.1
in the color section of Rossing's text.
When the light is reflected from a low index incident/high index
transmission interface, there is a phase change of pi radians (or 180
degrees).
For the soap film, the reflection from the second interface (surface) is
high index/low index and there is no phase change upon reflection. As the
film get very thin (much less than a wavelength of light) the two
reflections interfere destructively for all wavelengths. For thicker
regions of the film, the light interferes constructively when the path of
the light in the film is 1/2 a wavelength corresponding to a film
thickness of (1/4)λ. Note that the λ here is the λ in the
film, not the λ
in air. You do know that they are not the same don't you? What is the wavelength (λ)
in
the film?
Earlier we refered to the reflection for light going from low index
to high index as a "hard" reflection and for light going from high index to
low index as a "soft" reflection. In that context a hard reflection means
a phase shift of 180°, and a soft reflection means there is 0°
phase shift due to the reflection.
If the thin film is on a surface such as glass so that the reflection
at the second surface is also low/high, then there will be a phase change
at both interfaces. (That is, both reflections are "hard.") Then for a film
l/4 wavelength thick, the two reflected beams will be
exactly out of phase and interfere destructively for that particular
wavelength. Thus such a coating is an antireflection coating.
Coatings of this type are routinely used to coat camera and binocular
lenses.
In the video demonstration you saw thin film color filters known as dichroic filters.
Their color is not due to absorption but rather to interference. One color of
light is transmitted and the complementary color is reflected. Filters
of this type are used in television cameras to separate the primary colors (red,
green, and blue) and produce separate images of each on the corresponding tube or
semiconductor detector. Interference layers are used also to make antireflection coatings and to make
dielectric mirrors. (Dielectric refers to the material used being a nonconductor or non-metal.) By using
multiple quarter-wave layers, it is possible to improve on the antireflection and reflection properties of
the thin-film layer. Mirrors can be made that are tailored to specific requirements. For example, the
dielectric mirrors use in the projection lamps reflect the visible light but permit the infrared (heat)
radiation to pass through.
The diffraction by a single slit may be thought of as the interference of
waves from all across the width of the slit. In that case there will be a
bright central maxima surrounded by dimmer secondary maxima. The first minimum
occurs at a position y on the screen given by
1st minimum at y = (λ)(L/b)
where L is the distance to the screen and b is the width of the slit. Notice
that the distance y increases as b gets smaller. Most of the light intensity
is in the central maximum as shown in Figure 5.9 on p. 115 of your text.
There is a similar behavior in the diffraction of light by a
circular aperture. A central circular disk is seen in the light passed
by the aperture and there is a dark ring surrounded by alternating light
and dark rings about the central spot. See the illustrations in your text on p. 119. The angular
diameter of the central disk as measured from the center of the aperture
to the screen on which the pattern is displayed is determined by the
diameter of the aperture and the wavelength of the light (analogous to
the slit pattern depending on the width of the slit). When two objects
(light sources, etc.) are imaged through a circular aperture of diameter
D, they can only be seen as two (that is they are resolved) if the
separation of their images is greater than the radius of their central
diffraction disk. (That is, the maximum of one should fall on the first minimum of the other.
Thus the minimum angular separation of two objects to
be resolved is
minimum angle in radians = 1.22*(wavelength)/(diameter of lens)
or
θmin = 1.22λ/D (angle in radians)
The angle can also be found in degrees from
minimum angle in degrees = 70* (wavelength)/(diameter of lens)
or
θmin = 70λ/D (angle in degrees)
This effect occurs with a pinhole as well as with a lens. When making a
pinhole camera, reducing the diameter of the hole increases the sharpeness
of the image until the hole gets very small and then it further reduction
of the hole diameter causes the image to get blurry again.
Note that this minimum angle is called the Rayleigh angle in some texts.
This statement of the minimum angle is known as the Rayleigh
criterion for determining resolution. Be sure to study Figures 5.13 on p. 119.
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Last Modified:10/17/05