Physical Optics: Interference and Diffraction
Answers to Even-numbered Conceptual Questions
If the slit spacing,
, were less than the wavelength,
, the condition for a bright fringe
(Equation 28-1) could be satisfied only for the central bright fringe (
= 0). For nonzero
there are no solutions, because sin
cannot be greater than one. In addition,
Equation 28-2 shows that if
is greater than
/2, though still less than
, there will be
only one dark fringe on either side of the central bright fringe.
is less than
dark fringes will be observed.
The locations of bright and dark fringes depends on the wavelength of light.
white light is used in a two-slit experiment, each bright fringe will show some separation
into colors, giving a “rainbow” effect.
Submerging the two-slit experiment in water would reduce the wavelength of the light
= 1.33 is the index of refraction of water.
Therefore, the angles to
all the bright fringes would be reduced, as we can see from Equation 28-1.
It follows that
the two-slit pattern of bright fringes would be more tightly spaced in this case.
The soap film in the photograph is thinnest near the top (as one might expect) because in
that region the film appears black.
Specifically, light reflected from the front surface of
the film has its phase changed by 180
; light that reflects from the back surface of the film
has no change in phase.
Therefore, light from the front and back surfaces of the film will
undergo destructive interference as the path length between the surfaces goes to zero.
This is why the top of the film, where the film is thinnest, appears black in the photo.
One possible reason is that one of the films may have an index of refraction greater than
that of glass, whereas the other may have an index of refraction that is less than that of
If this is the case, the phase change in reflection from the film-glass interface will
be different for the two films.
This, in turn, would result in different colors appearing in
the reflected light.
Light reflected from the top of the film has a phase change of 180
; light reflected from
the film-water surface also has a phase change of 180
, since the film’s index of refraction
is less than that of water.
It follows that the film appears bright (constructive interference)
where the film’s thickness goes to zero.
The location of dark fringes in a single-slit diffraction pattern is given by Equation 28-12.
Notice that if
is decreased, the angle
must increase to compensate, and to maintain the
This is why the dark fringes move outward.
As the angle
= ±1, all of the dark fringes have moved outward to infinity.
Clearly, this occurs in
Equation 28-12 when the width
is equal to the wavelength,